6.–10. oktober 2025 l 6–10 October 2025 Koper, Slovenia IS 2025 INFORMACIJSKA DRUZBA ˇ INFORMATION SOCIETY Srednjeevropska konferenca o uporabnem teoretičnem računalništvu in informatiki (MATCOS) Middle-European Conference Zbornik 28. mednarodne on Applied Theoretical multikonference Computer Science Zvezek N (MATCOS) Proceedings of the 28th International Multiconference Volume N Uredniki l Editors: Andrej Brodnik, Gábor Galambos, Rok Požar Zbornik 28. mednarodne multikonference INFORMACIJSKA DRUŽBA – IS 2025 Zvezek N Proceedings of the 28th International Multiconference INFORMATION SOCIETY – IS 2025 Volume N Srednjeevropska konferenca o uporabnem teoretičnem računalništvu in informatiki (MATCOS) Middle-European Conference on Applied Theoretical Computer Science (MATCOS) Uredniki / Editors Andrej Brodnik, Gábor Galambos, Rok Požar http://is.ijs.si 9.–10. oktober 2025 / 9–10 October 2025 Koper, Slovenia Uredniki: Andrej Brodnik Univerza na Primorskem, Koper in Univerza v Ljubljani, Ljubljana Gábor Galambos Univerza v Szegedu, Szeged Rok Požar Univerza na Primorskem, Koper Založnik: Institut »Jožef Stefan«, Ljubljana Priprava zbornika: Mitja Lasič, Vesna Lasič, Lana Zemljak Oblikovanje naslovnice: Vesna Lasič, uporabljena slika iz Pixabay Dostop do e-publikacije: http://library.ijs.si/Stacks/Proceedings/InformationSociety Ljubljana, oktober 2025 Informacijska družba ISSN 2630-371X DOI: https://doi.org/10.70314/is.2025.matcos Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici v Ljubljani COBISS.SI-ID 255599107 ISBN 978-961-264-332-4 (PDF) PREDGOVOR MULTIKONFERENCI INFORMACIJSKA DRUŽBA 2025 28. mednarodna multikonferenca Informacijska družba se odvija v času izjemne rasti umetne inteligence, njenih aplikacij in vplivov na človeštvo. Vsako leto vstopamo v novo dobo, v kateri generativna umetna inteligenca ter drugi inovativni pristopi oblikujejo poti k superinteligenci in singularnosti, ki bosta krojili prihodnost človeške civilizacije. Naša konferenca je tako hkrati tradicionalna znanstvena in akademsko odprta, pa tudi inkubator novih, pogumnih idej in pogledov. Letošnja konferenca poleg umetne inteligence vključuje tudi razprave o perečih temah današnjega časa: ohranjanje okolja, demografski izzivi, zdravstvo in preobrazba družbenih struktur. Razvoj UI ponuja rešitve za številne sodobne izzive, kar poudarja pomen sodelovanja med raziskovalci, strokovnjaki in odločevalci pri oblikovanju trajnostnih strategij. Zavedamo se, da živimo v obdobju velikih sprememb, kjer je ključno, da z inovativnimi pristopi in poglobljenim znanjem ustvarimo informacijsko družbo, ki bo varna, vključujoča in trajnostna. V okviru multikonference smo letos združili dvanajst vsebinsko raznolikih srečanj, ki odražajo širino in globino informacijskih ved: od umetne inteligence v zdravstvu, demografskih in družinskih analiz, digitalne preobrazbe zdravstvene nege ter digitalne vključenosti v informacijski družbi, do raziskav na področju kognitivne znanosti, zdrave dolgoživosti ter vzgoje in izobraževanja v informacijski družbi. Pridružujejo se konference o legendah računalništva in informatike, prenosu tehnologij, mitih in resnicah o varovanju okolja, odkrivanju znanja in podatkovnih skladiščih ter seveda Slovenska konferenca o umetni inteligenci. Poleg referatov bodo okrogle mize in delavnice omogočile poglobljeno izmenjavo mnenj, ki bo pomembno prispevala k oblikovanju prihodnje informacijske družbe. »Legende računalništva in informatike« predstavljajo domači »Hall of Fame« za izjemne posameznike s tega področja. Še naprej bomo spodbujali raziskovanje in razvoj, odličnost in sodelovanje; razširjeni referati bodo objavljeni v reviji Informatica, s podporo dolgoletne tradicije in v sodelovanju z akademskimi institucijami ter strokovnimi združenji, kot so ACM Slovenija, SLAIS, Slovensko društvo Informatika in Inženirska akademija Slovenije. Vsako leto izberemo najbolj izstopajoče dosežke. Letos je nagrado Michie-Turing za izjemen življenjski prispevek k razvoju in promociji informacijske družbe prejel Niko Schlamberger, priznanje za raziskovalni dosežek leta pa Tome Eftimov. »Informacijsko limono« za najmanj primerno informacijsko tematiko je prejela odsotnost obveznega pouka računalništva v osnovnih šolah. »Informacijsko jagodo« za najboljši sistem ali storitev v letih 2024/2025 pa so prejeli Marko Robnik Šikonja, Domen Vreš in Simon Krek s skupino za slovenski veliki jezikovni model GAMS. Iskrene čestitke vsem nagrajencem! Naša vizija ostaja jasna: prepoznati, izkoristiti in oblikovati priložnosti, ki jih prinaša digitalna preobrazba, ter ustvariti informacijsko družbo, ki koristi vsem njenim članom. Vsem sodelujočim se zahvaljujemo za njihov prispevek — veseli nas, da bomo skupaj oblikovali prihodnje dosežke, ki jih bo soustvarjala ta konferenca. Mojca Ciglarič, predsednica programskega odbora Matjaž Gams, predsednik organizacijskega odbora i FOREWORD TO THE MULTICONFERENCE INFORMATION SOCIETY 2025 The 28th International Multiconference on the Information Society takes place at a time of remarkable growth in artificial intelligence, its applications, and its impact on humanity. Each year we enter a new era in which generative AI and other innovative approaches shape the path toward superintelligence and singularity — phenomena that will shape the future of human civilization. The conference is both a traditional scientific forum and an academically open incubator for new, bold ideas and perspectives. In addition to artificial intelligence, this year’s conference addresses other pressing issues of our time: environmental preservation, demographic challenges, healthcare, and the transformation of social structures. The rapid development of AI offers potential solutions to many of today’s challenges and highlights the importance of collaboration among researchers, experts, and policymakers in designing sustainable strategies. We are acutely aware that we live in an era of profound change, where innovative approaches and deep knowledge are essential to creating an information society that is safe, inclusive, and sustainable. This year’s multiconference brings together twelve thematically diverse meetings reflecting the breadth and depth of the information sciences: from artificial intelligence in healthcare, demographic and family studies, and the digital transformation of nursing and digital inclusion, to research in cognitive science, healthy longevity, and education in the information society. Additional conferences include Legends of Computing and Informatics, Technology Transfer, Myths and Truths of Environmental Protection, Knowledge Discovery and Data Warehouses, and, of course, the Slovenian Conference on Artificial Intelligence. Alongside scientific papers, round tables and workshops will provide opportunities for in-depth exchanges of views, making an important contribution to shaping the future information society. Legends of Computing and Informatics serves as a national »Hall of Fame« honoring outstanding individuals in the field. We will continue to promote research and development, excellence, and collaboration. Extended papers will be published in the journal Informatica, supported by a long-standing tradition and in cooperation with academic institutions and professional associations such as ACM Slovenia, SLAIS, the Slovenian Society Informatika, and the Slovenian Academy of Engineering. Each year we recognize the most distinguished achievements. In 2025, the Michie-Turing Award for lifetime contribution to the development and promotion of the information society was awarded to Niko Schlamberger, while the Award for Research Achievement of the Year went to Tome Eftimov. The »Information Lemon« for the least appropriate information-related topic was awarded to the absence of compulsory computer science education in primary schools. The »Information Strawberry« for the best system or service in 2024/2025 was awarded to Marko Robnik Šikonja, Domen Vreš and Simon Krek together with their team, for developing the Slovenian large language model GAMS. We extend our warmest congratulations to all awardees. Our vision remains clear: to identify, seize, and shape the opportunities offered by digital transformation, and to create an information society that benefits all its members. We sincerely thank all participants for their contributions and look forward to jointly shaping the future achievements that this conference will help bring about. Mojca Ciglarič, Chair of the Program Committee Matjaž Gams, Chair of the Organizing Committee iiii KONFERENČNI ODBORI CONFERENCE COMMITTEES International Programme Committee Organizing Committee Vladimir Bajic, South Africa Matjaž Gams, chair Heiner Benking, Germany Mitja Luštrek Se Woo Cheon, South Korea Lana Zemljak Howie Firth, UK Vesna Koricki Olga Fomichova, Russia Mitja Lasič Vladimir Fomichov, Russia Blaž Mahnič Vesna Hljuz Dobric, Croatia Alfred Inselberg, Israel Jay Liebowitz, USA Huan Liu, Singapore Henz Martin, Germany Marcin Paprzycki, USA Claude Sammut, Australia Jiri Wiedermann, Czech Republic Xindong Wu, USA Yiming Ye, USA Ning Zhong, USA Wray Buntine, Australia Bezalel Gavish, USA Gal A. Kaminka, Israel Mike Bain, Australia Michela Milano, Italy Derong Liu, Chicago, USA Toby Walsh, Australia Sergio Campos-Cordobes, Spain Shabnam Farahmand, Finland Sergio Crovella, Italy Programme Committee Mojca Ciglarič, chair Marjan Heričko Boštjan Vilfan Bojan Orel Borka Jerman Blažič Džonova Baldomir Zajc Franc Solina Gorazd Kandus Blaž Zupan Viljan Mahnič Urban Kordeš Boris Žemva Cene Bavec Marjan Krisper Leon Žlajpah Tomaž Kalin Andrej Kuščer Niko Zimic Jozsef Györkös Jadran Lenarčič Rok Piltaver Tadej Bajd Borut Likar Toma Strle Jaroslav Berce Janez Malačič Tine Kolenik Mojca Bernik Olga Markič Franci Pivec Marko Bohanec Dunja Mladenič Uroš Rajkovič Ivan Bratko Franc Novak Borut Batagelj Andrej Brodnik Vladislav Rajkovič Tomaž Ogrin Dušan Caf Grega Repovš Aleš Ude Saša Divjak Ivan Rozman Bojan Blažica Tomaž Erjavec Niko Schlamberger Matjaž Kljun Bogdan Filipič Gašper Slapničar Robert Blatnik Andrej Gams Stanko Strmčnik Erik Dovgan Matjaž Gams Jurij Šilc Špela Stres Mitja Luštrek Jurij Tasič Anton Gradišek Marko Grobelnik Denis Trček Nikola Guid Andrej Ule iii iv KAZALO / TABLE OF CONTENTS Srednjeevropska konferenca o uporabnem teoretičnem računalništvu in informatiki (MATCOS) / Middle-European Conference on Applied Theoretical Computer Science (MATCOS) ....................... 1 PREDGOVOR / FOREWORD ............................................................................................................................... 3 PROGRAMSKI ODBORI / PROGRAMME COMMITTEES ............................................................................... 5 Merging Operations in the Open-Shop Scheduling Problem / Baldouski Daniil, Dávid Balázs, Krész Miklós .... 7 Flexibility vs. efficiency: a study in sawmill scheduling / Kebelei Csaba, Hegyháti Máté ................................. 11 Implementation of a Vehicle and Driver Scheduling Model: a Case Study / Árgilán Viktor, Békési József, Galambos Gábor, Papp Imre ............................................................................................................................ 15 ALGatorGraph: A Java Library for Graph Generation and Manipulation within the ALGator System / Hren Boštjan, Dobravec Tomaž ................................................................................................................................ 19 Engineering CSFLOC: A Subsumption-Driven Clause-Counting SAT Solver / Kusper Gábor ........................ 23 Non-redundant Systems of Independence Atoms in Relational Databases / Alland Lucas, Sali Attila, Wu Nicole .......................................................................................................................................................................... 27 Scrambler Automaton Block Cipher for IoT Devices / Dömösi Pál, Horváth Géza ............................................ 30 Automata for context-free trace languages and permutation languages / Nagy Benedek .................................... 35 Towards a Category-Theoretic Informatics Model of PSPP Linkages in Biomaterials / Tahalea Sylvert Prian, Krész Miklós .................................................................................................................................................... 39 Cost-Sensitive Overview of Model Ensembling for Machine-Generated Text Detection / Kiss Mihály, Berend Gábor ................................................................................................................................................................ 44 Hybrid Reinforcement Learning Enhanced Genetic Algorithm for the Capacitated Vehicle Routing Problem with Split Deliveries and Heterogeneous Fleet / Dabbous Ahmed, Bóta András ........................................... 48 Empiric results on the achievable performance gains by the inclusion of instance-specific information in a scheduling optimizer / Hegyháti Máté ............................................................................................................ 52 Bi-Level Routing and Scheduling / Quilliot Alain, Toussaint Hélène ................................................................. 56 Adiscrete event simulation model for analyzing the wood waste reverse supply chain / Kovačević Nikola, Tavzes Črtomir, Dávid Balázs.......................................................................................................................... 60 Robust (re)Design of Material Flow in Circular Networks – a Scientific Approach / Szaller Ádám, Dávid Balázs, Egri Péter, Krész Miklós, Váncza József ............................................................................................. 64 Harvest plan generation in precision agriculture / Horvat Štefan, Strnad Damjan, Mongus Domen, Brumen Matej ................................................................................................................................................................ 68 Reducing #SAT to k-clique enumeration / Szabó Sándor, Zaválnij Bogdán ....................................................... 72 Modularity aware graph clustering for exploratory tasks with a case study of the biomass supply chain / Tahalea Sylvert Prian, Kawa Arkadiusz, Dávid Balázs ................................................................................... 75 ASynthetic Multi-View Tracking and 3D Pose Dataset for Automated Airport Visual Surveillance / Mansour Ahmed, Beleznai Csaba, Oberweger Fabio F., Widhalm Verena, Kirillova Nadezda, Possegger Horst ......... 80 Sphere Target-Based Point Cloud Registration in a Railway Safety Application / Podgorelec David, Lukač Luka, Pečnik Sašo, Repnik Blaž, Žalik Borut .................................................................................................. 84 Indeks avtorjev / Author index ................................................................................................................... 89 v vi Zbornik 28. mednarodne multikonference INFORMACIJSKA DRUŽBA – IS 2025 Zvezek N Proceedings of the 28th International Multiconference INFORMATION SOCIETY – IS 2025 Volume N Srednjeevropska konferenca o uporabnem teoretičnem računalništvu in informatiki (MATCOS) Middle-European Conference on Applied Theoretical Computer Science (MATCOS) Uredniki / Editors Andrej Brodnik, Gábor Galambos, Rok Požar http://is.ijs.si 9.–10. oktober 2025 / 9–10 October 2025 Koper, Slovenia 1 2 PREDGOVOR To je že peti MATCOS – konferenca, ki povezuje dva na videz nepovezljiva pojma, aplikativnost in teorijo, v konferenco o uporabnem teoretičnem računalništvu. Ko smo leta 2013 organizirali prvo konferenco MATCOS, smo lahko le sanjali o ustvarjanju tradicije. Naše prejšnje konference so se osredotočale na gradnjo: poskušali smo ustvariti novo konferenčno prizorišče v Srednji Evropi in zelo trdo smo delali, da bi to dosegli. Številna mesta poskušajo z organizacijo konferenc na različnih področjih računalništva in/ali matematike, vendar ne uspe vsem uresničiti njihovih načrtov. Organizacija konference je težko delo, a zahvaljujoč neutrudnim prizadevanjem članov OO in PO ter vseh, ki so uporabili svoje povezave, smo tukaj. Po zaključenem recenzentskem postopku se je PO odločil sprejeti 20 rednih prispevkov, ki so tudi vključeni v ta zbornik. V skladu z našimi prejšnjimi prizadevanji, želimo tudi letos dati priložnost mladim raziskovalcem. Zato smo posebno pozornost namenili prispevkom, ki so jih predložili doktorski študenti, in v nekaterih primerih nekaj od njih vključili med redne predstavitve. 14 kratkih predstavitev, ki so bile sprejete, ponuja priložnost za nadaljnjo razširitev spektra raziskav predstavljenih na MATCOS. V preteklih letih smo na plenarna predavanja na konferencah MATCOS povabili predavatelje, ki so vodilni strokovnjaki na svojih področjih. Letošnja konferenca ni izjema: Michel Bierlaire, profesor na École Polytechnique Fédérale de Lausanne (EPFL), se je odzval našemu povabilu. Njegovo raziskovalno področje – raziskave prometa – se dobro ujema z okvirom MATCOS, njegova prisotnost pa ponuja dobro priložnost za vzpostavitev in poglobitev osebnih stikov. Vse udeležence MATCOS-25 lepo pozdravljamo in upamo, da boste uživali v teh dveh dneh v Kopru. Spoznajte mesto, saj je Koper eden od diamantov Istre. Prepričani smo, da bodo prihajajoči dnevi prispevali k nadaljnjemu razvoju obstoječih odnosov in upamo, da se bodo s pomočjo konference MATCOS oblikovale tudi nove strokovne skupine. V imenu organizatorjev Andrej Brodnik in Gábor Galambos Sopredsednika 3 FOREWORD This is already the fifth MATCOS – the conference that connects two seemingly unconnectable terms application and theory into a conference on applied theoretical computer science. When we started to organize the first MATCOS conference back in 2013, we could only dream of creating a tradition. Our previous conferences focused on building: we tried to create a new conference venue in Central Europe, and we worked very hard to achieve this. Many cities are experimenting with organizing conferences in various fields of computer science and/or mathematics, but not everyone can manage to realize their plans. Organizing a conference is a hard work, but thanks to the tireless efforts of OC and PC members, and everyone who used their connections, here we are. As a result of the review process, the PC accepted 20 regular papers, and these are included in the Proceedings. In keeping with our previous efforts, we want to give young researchers a chance this year as well, so we gave special consideration to the papers submitted by PhD students and, in some cases, included a few of them among the regular talks. The 14 short talks that were accepted provide an opportunity to further broaden the spectrum of research presented at MATCOS. In previous years, we have invited speakers who are leading experts in their fields to give plenary presentations at MATCOS conferences. This year's conference is no exception: Michel Bierlaire, professor at the École Polytechnique Fédérale de Lausanne (EPFL), has accepted our invitation. His research field — transport research — fits well within the scope of MATCOS, and his presence provides a good opportunity to establish and deepen personal relationships. A warm welcome to all MATCOS-25 participants, and we hope you enjoy these two days in Koper. Get to know the city, because Koper is one of the diamond of Istria. We are confident that the coming days will contribute to the further development of existing relationships, and we hope that new professional groups will also be formed with the help of the MATCOS conference. On behalf of the organizers Andrej Brodnik and Gábor Galambos Co-chairs 4 PROGRAMSKI ODBOR / PROGRAMME COMMITTEE Andrej Brodnik (Koper, Ljubljana, Slovenia) co-chair Bo Chen (Warwick, UK) Gábor Galambos (Szeged, Hungary) co-chair Kathrin Hanauer (Vienna, Austria) Gabriel Istrate (Bucharest, Romania) Miklós Krész (Szeged, Hungary and Koper, Slovenia) Silvano Martello (Bologna, Italy) Andrzej Mizera (Warsaw, Poland) Benedek Nagy (Famagusta, Cyprus, Turkey) Bengt J. Nilsson (Malmö, Sweeden) Ulrich Pferschy (Graz, Austria) Gerhard Reinelt (Heidelberg, Germany) Aleksi Saarela (Turku, Finland( Attila Sali (Budapest, Hungary) Yllka Velaj (Vienna, Austria) Borut Žalik (Maribor, Slovenia) 5 6 Merging Operations in the Open-Shop Scheduling Problem Daniil Baldouski Balázs Dávid Miklós Krész daniil.baldouski@iam.upr.si balazs.david@innorenew.eu miklos.kresz@innorenew.eu University of Primorska, IAM InnoRenew CoE InnoRenew CoE Koper, Slovenia Izola, Slovenia Izola, Slovenia University of Primorska, IAM and University of Primorska, IAM and FAMNIT FAMNIT Koper, Slovenia Koper, Slovenia University of Szeged Szeged, Hungary ABSTRACT • The time horizon of the problem is represented by the In this work we consider the open-shop scheduling problem with discretized set of consecutive time units 𝑇 . operation batching (OSSP-OB), which extends classical open- • A finite set 𝐽 of jobs. For each job 𝑗 ∈ 𝐽 there is a time shop scheduling by allowing operations of the same category window [𝑎𝑗, 𝑏 𝑗 ] within the time horizon of the problem to be processed simultaneously on a machine. To address this for which the operations of this job should be analyzed problem, we develop an exact procedure for constructing optimal and approved. batches with the corresponding machine scheduling by devel- • A finite set 𝑂 of operations associated with jobs. For each oping a mixed-integer linear programming (MILP) model. The job 𝑗 ∈ 𝐽 , a set 𝑂𝑗 ⊆ 𝑂 of operations associated with the effectiveness of the method is evaluated on a dedicated bench- job. mark instance set. • A finite set 𝐶 of operation categories associated with operations. Each operation 𝑜 ∈ 𝑂 has a category 𝐶 (𝑜) KEYWORDS and there are no two operations of the same category that belong to the same job. The time it takes to perform the job scheduling, batch scheduling, open-shop scheduling, mixed- operation is the same within the operation category and integer linear programming is then defined for each category 𝑐 ∈ 𝐶 as ℓ(𝑐) (the length 1 INTRODUCTION category is denoted by of the category). The set of all operations of the same 𝑂 𝑐 𝑐 𝐶 ( ) for each category ∈. We consider a scheduling problem that combines ideas from • A finite set 𝑀 of machines. For each machine 𝑚 ∈ 𝑀, a open-shop and batch scheduling. Each job consists of several set 𝐶𝑚 ⊆ 𝐶 of operation categories that can be carried out operations, each associated with a category and executable on on the machine. one of the suitable machines for a specified processing time. • Operations can be performed in batches if they belong to Operations within a job can be processed in arbitrary order, so the the same category. This will be referred to as merging problem environment belongs to the class of open-shop scheduling of operations. Each merged operation 𝑠 ∈ 𝑆 is a set of (OSSP) [3, 4, 1]. operations, that is, 𝑠 ⊆ 𝑂 such that 𝑠 contains operations In addition, we allow multiple operations of the same category of the same category. The set of all merged operations to be executed simultaneously on a machine. This grouping pro- can be understood as a set 𝑆 ⊆ P (𝑂) that contains all the cedure, which we call merging, is related to the notion of batching possible valid merged operations. in batch scheduling [5, 2, 7], but differs as it applies to operations Additional parameters can include, but are not limited to: rather than jobs. The resulting formulation, referred to as the open-shop sched- • capacity parameters for the number of operations for each uling problem with operation batching (OSSP-OB), extends the machine, classical OSSP by incorporating merging of operations and cap- • capacity of the number of merged jobs for each category, tures a wider range of scheduling scenarios. The OSSP can then • time window for which the operations of the job should be viewed as a special case of OSSP-OB in which a unique cate- be analyzed, gory is provided for every operation, thus leading OSSP-OB to • internal deadline of the job, be an NP-hard problem in the general case. • periods of activity/inactivity, • time to merge operations (e.g. constant, flexible, propor- 2 PROBLEM DESCRIPTION tional to the execution time, proportional to the number of merged operations, and more) The environment of the open-shop scheduling problem with the merging of operations (operations batching) is given by the sets • precedence relations between the operations within the same job, of jobs, operations, and machines, with various parameters for each of the sets. Formally, OSSP-OB environment consists of: • precedence relations between jobs. In this work, we consider two additional parameters that are Permission to make digital or hard copies of part or all of this work for personal of high importance in the use-case of laboratory experiments: or classroom use is granted without fee provided that copies are not made or internal deadlines and activity periods. distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this The internal deadline (𝑑𝑗 ) can be defined as 𝑘 time units be- work must be honored. For all other uses, contact the owner/author(s). 𝑏 fore 𝑗 , which is the desired internal target for operations to Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia be analyzed and approved, which then gives some time ( 𝑘 time © 2025 Copyright held by the owner/author(s). units) for corrections in the case of exceptions (failures). A good 7 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Baldouski et al. example of a merging parameter would be a limit on the number • 𝑡𝐶 (𝑐)- time required to perform the operations of cate- of operations that can be merged (𝑚(𝐶)) for each category 𝑐 ∈ 𝐶 . gory 𝑐. In the context of QA laboratory experiments, personnel sched- • 𝑎 𝑡 𝐽 ( 𝑗)- the start of the time interval for which the uling plays an important role, since most tasks (such as machine operations of job 𝑗 should be analyzed. preparation, machine loading, validation of the results, and more) 𝑏 • 𝑡 𝐽( 𝑗)- the end of the time interval for which the oper- are performed manually, each requiring a specific employee qual- ations of job 𝑗 should be analyzed. ification to handle it. In this work, we consider personnel sched- • 𝑑 𝑡 𝐽 ( 𝑗 )- the internal deadline of the time interval for uling in a simplified format, assuming that all manual tasks are which the operations of job 𝑗 should be analyzed. performed instantaneously and with infinite personnel capacity. • 𝐴- set of activity periods, 𝐴 ⊆ 𝑇 . Let us call periods of inactivity all the gaps in the time horizon (3) Auxiliary parameters. A set of derived parameters fol- (e.g., weekends, national holidays, other special events) during lows, simplifying the connection between the set of merged which machines are unavailable and none of the tasks can be operations and: performed. Let us then consider an activity period to be any • Categories: maximal interval of time units that does not contain periods of – 𝑎𝑆𝐶 (𝑠, 𝑐 ) = 1 if 𝑐 = 𝑎𝑂𝐶 (𝑜) for any 𝑜 ∈ 𝑠, 0 otherwise, inactivity. for a merged operation 𝑠. An example of the problem (Fig. 1) with a feasible solution – Let 𝑠 𝑐 ∈ 𝐶 be the category of a merged operation 𝑠, (Fig. 2) is demonstrated. In this example, we are given 3 jobs 𝑠 that is 𝑎𝑆𝐶 ( 𝑠, 𝑐) = 1. (𝐽 = {𝐽 1, 𝐽 2, 𝐽 3}, sorted by the earliest allowed start time), each • Jobs: of them containing 3 − 4 operations. The time windows of the – 𝑎𝑆 𝐽 (𝑠, 𝑗) = 1 if 𝑗 ∈ {𝑎𝑂 𝐽 (𝑜) : 𝑜 ∈ 𝑠} and 0 other- jobs are given ([𝐴( 𝑗 ), 𝐵( 𝑗 )] for each job 𝑗 ∈ 𝐽), as well as the wise, for a merged operation 𝑠. That is, if the merged lengths of the operations (categories). The colors of the opera- operation 𝑠 and the job 𝑗 are associated with each tions represent their categories and there are 6 categories in total other. 𝐶 𝐶 , . . . , 𝐶 𝑂 𝐽 𝑂 , 𝑂 , 𝑂 = { 1 6 } . For example, ( 2 ) = { 1 23} such that • Machines: 𝐶 𝑂 𝐶 𝑎𝑆 𝑀 𝑠, 𝑚 𝑐 𝐶 ( 3 ) = 1. – ( ) = 1 if there is at least one category ∈ There are 3 machines (𝑀 = {𝑀1, 𝑀2, 𝑀3}), each of them such that 𝑎𝑆𝐶(𝑠, 𝑐) = 1 and 𝑎𝑀𝐶 (𝑚, 𝑐) = 1, 0 other- corresponds to a set of categories, of which operations are allowed wise, for merged operation 𝑠 and machine 𝑚. to be processed. In this example, 𝐶𝑀 1 = {𝐶1, . . . , 𝐶4}, while – Let set 𝑠 𝑀 be the set of machines compatible with the 𝐶 𝐶 𝐶 , . . . , 𝐶 𝑠 𝑀 2 = 𝑀 3 = { 4 6 } . The example does not dive deep into merged operation, that is, for every 𝑠 ∈ 𝑆: all the possible parameters of the problem, assuming that merging 𝑀 𝑚 𝑚 𝑀 , 𝑎𝑆 𝑀 𝑠, 𝑚 . is allowed in case operations belong to the same category, while 𝑠 = { | ∈ ( ) = 1} satisfying the time constraints. Additionally, we introduce a big M parameter based on A feasible solution (Fig. 2) is given in the form of performed the time horizon 𝑇 : merges, (merged) operations-machines assignment, with the (merged) operations timeline. There are 4 potential merges for cat- M = 2 · |𝑇 | + 1. egories 𝐶1, 𝐶2, 𝐶4 and 𝐶5, representing different types of merges that can occur. For example, merging of category 𝐶 1 is not al- 3.2 Variables lowed due to time limitation (ℓ (𝐶1) > 𝐵 (𝐽 1) − 𝐴( 𝐽 2)). Because of For each 𝑠 ∈ 𝑆, 𝑗 ∈ 𝐽 , 𝑜 ∈ 𝑂, 𝑚 ∈ 𝑀, 𝑐 ∈ 𝐶, and 𝜏 ∈ 𝑇 , we this, the schedule of machine 𝑀1 becomes tight, making the merg- establish the following variables. 𝐶 (1) Assignment variables: 4 and 𝐶 5 are possible but not necessary and would be performed • ing of operations of category 𝐶2 necessary. Merges of categories (or not) based on the specifics of the objective function. 𝑥 𝑠,𝑚,𝜏 𝑠 In general, the goal of this work is to find the most efficient machine 𝑚 in time unit 𝜏. • = 1 if and only if merged operation is active on scheduling of operations to machines given the merging con- 𝑦𝑠,𝜏 𝑠 = 1 if and only if the merged operation starts in straints, with the objectives to consider including tardiness, num- the time unit 𝜏. • 𝑧 ber of batches, flowtime, makespan, and overall costs. 𝑠,𝑚 = 1 if and only if the merged operation 𝑠 and machine 𝑚 are assigned to each other during the time 3 horizon 𝑇 . SOLUTION METHODOLOGY • 𝑟𝑠 = 1 if and only if the merged operation 𝑠 is active In this work, we formulate the OSSP-OB as a mixed-integer linear during the time horizon 𝑇 . programming (MILP) model that integrates both the merging of (2) Auxiliary variables. A set of helpful variables to simplify operations and the scheduling of merged operations. the model formulation, grouping the merged operations parameters together with the decision variable 𝑟𝑠 that 3.1 Problem data represents the activity status of a merged operation: For each 𝑠 ∈ 𝑆 , 𝑗 ∈ 𝐽 , 𝑜 ∈ 𝑂, 𝑚 ∈ 𝑀, 𝑐 ∈ 𝐶, and 𝜏 ∈ 𝑇 , we • 𝑐 𝑟 𝑠,𝑐 = 𝑟𝑠 · 𝑎𝑆𝐶 (𝑠, 𝑐) for merged operation 𝑠 and category establish the following parameters. 𝑐. • 𝑗 = · ( ) for merged operation and job . (1) 𝑟 𝑟𝑠 𝑎𝑆 𝐽 𝑠, 𝑗 𝑠 𝑗 Assignment parameters: 𝑠, 𝑗 • • 𝑚 𝑟 𝑟 𝑎𝑂 𝐽 ( 𝑜 )- the index of the job associated with the opera-𝑠,𝑚 =𝑠 · 𝑎𝑆 𝑀 (𝑠, 𝑚) for merged operation 𝑠 and ma- • tion chine 𝑚. 𝑜 . 𝑐 𝑗 •𝑐 𝑗 𝑟 𝑎𝑂𝐶 = ( 𝑟 𝑜 · )- the index of the category of the operation 𝑟 𝑠 𝑐 𝑜 . 𝑠,𝑐, 𝑗 𝑠,𝑐 𝑠, 𝑗 for merged operation , category and • 𝑎𝑀𝐶(𝑚, 𝑐) = job 1 if and only if operations of category 𝑗 𝑐 . can be carried out on the machine 𝑚. • 𝑗 = · for merged operation , machine 𝑘𝑠,𝑚, 𝑗 ,𝜏 𝑟 𝑥𝑠,𝑚,𝜏 𝑠 𝑠, 𝑗 (2) Processing time parameters: 𝑚, category 𝑐 and time unit 𝜏. 8 Merging Operations in the Open-Shop Scheduling Problem Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Figure 1: Example of OSSP-OB with 3 jobs and 3 − 4 operations per job. Figure 2: Example of an OSSP-OB solution, demonstrating operations to machines scheduling. 3.3 Constraints • Exactly one machine is assigned: (1) Auxiliary variables definitions. For each 𝑠 ∈ 𝑆, 𝑚 ∈ 𝑀, ∑︁ 𝑧 𝑟 = (10) 𝑐 𝐶 𝑗 𝐽 𝜏 𝑇 , ∈ , ∈ : ∈ 𝑠,𝑚 𝑠 • 𝑐 𝑟 𝑟 𝑠,𝑐 𝑚 𝑀 ∈ = 𝑠 · 𝑎𝑆𝐶 (𝑠, 𝑐 ) . • 𝑗 · ( ) = (4) Merged operations can not be scheduled on machines that . 𝑟 𝑟𝑠 𝑎𝑆 𝐽 𝑠, 𝑗 𝑠, 𝑗 do not share the corresponding category. For each merged • 𝑚 𝑟 𝑟 𝑠, 𝑚 𝑠 𝑠,𝑚 = · (). 𝑎𝑆 𝑀 • Linearization of 𝑟 : 𝑠 𝑐 𝑗 operation 𝑠 ∈ 𝑆 : • For each machine 𝑚 ∈ 𝑀 \ 𝑀 : 𝑐 𝑗 𝑐 ≤ (1) 𝑟 𝑟 𝑠,𝑐, 𝑗 𝑠,𝑐 ∑︁ 𝑥 𝑠,𝑚,𝜏 = 0 (11) 𝑐 𝑗 𝑗 ≤ (2) 𝑟 𝑟 ∈ 𝑠,𝑐, 𝑗 𝑠, 𝑗 𝜏 𝑇 𝑐 𝑗 𝑐 𝑗 𝑧𝑠,𝑚 = 0 (12) 𝑟 𝑟 𝑟 + − 1 ≤ (3) 𝑠,𝑐 𝑠, 𝑗 𝑠,𝑐, 𝑗 • 𝑠 For each machine ∈ : • 𝑚 𝑀 Linearization of : 𝑘 ∑︁ 𝑘 𝑗 𝑠 𝑥 𝑠,𝑚,𝜏 = 𝑟 𝑠 · 𝑡𝐶 ( ≤ 𝑠,𝑚, 𝑗 ,𝜏 𝑐 𝑟 ) (13) (4) 𝑠, 𝑗 𝜏 𝑇 ∈ 𝑘𝑠,𝑚, 𝑗 ,𝜏 𝑥𝑠,𝑚,𝜏 ≤ (5) 𝑗 (5) Merged operations should start and end within the corre- 𝑟 𝑥 𝑠,𝑚,𝜏 𝑘𝑠,𝑚, 𝑗 ,𝜏 𝑠 + − ≤ 𝑠, 𝑗 1 (6) sponding time interval. For each merged operation ∈ 𝑆, (2) At any time unit, merged operations and machines have job 𝑗 ∈ 𝐽 and category 𝑐 ∈ 𝐶: at most one unique schedule. For each time unit 𝜏 ∈ 𝑇 : ∑︁ 𝑗 𝑦𝑠,𝜏 𝑟 ≤ ( 1 − )M (14) • 𝑠, 𝑗 For each merged operation ∈ : 𝑠 𝑆 𝜏 𝑇 ,𝜏<𝑡 𝐽 𝑗 ( ) ∈ 𝑎 ∑︁ 𝑥 𝑐 𝑗 ≤ 1 (7) ≤ ( 1 (15) 𝑠,𝑚,𝜏 𝑦 − )M ∑︁ 𝑠,𝜏 𝑟 𝑠,𝑐, 𝑗 𝑚 𝑀 ∈ 𝜏 𝑇 ,𝜏>𝑡 𝐽 𝑗 𝑡𝐶 𝑐 ( ) − () ∈ 𝑏 • For each machine 𝑚 ∈ 𝑀: ∑︁ (6) Merged operations can be performed only on the dedicated 𝑥 𝑠,𝑚,𝜏 𝑠 ≤ 1 (8) machines. For each merged operation ∈ 𝑆 , machine 𝑠 𝑆 ∈ ∈ and category ∈ : 𝑚 𝑀 𝑐 𝐶 (3) For each merged operation 𝑠 ∈ 𝑆 to be scheduled (𝑟𝑠 = 1): • 𝑧𝑠,𝑚 𝑟𝑠,𝑚 There is exactly one starting time unit: ≤ 𝑚 (16) ∑︁ (7) Each merged operation ∈ occupies ( ) consecutive = (9) 𝑦𝑠,𝜏 𝑟𝑠 𝑠 𝑆 𝑡𝐶 𝑐 𝜏 𝑇 𝑠,𝑐 ∈ time units for a unique ∈ such that = 1. For each 𝑐 𝑐 𝐶 𝑟 9 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Baldouski et al. merged operation 𝑠 ∈ 𝑆, machine 𝑚 ∈ 𝑀, category 𝑐 ∈ 𝐶 N jobs N machines N categories Avg. Runtime (s) , time unit ′ ∈ and ∈ {0 1 ( ) − 1}: 3 2 2 1.11 𝜏 𝑇 𝜏 , , . . . , 𝑡𝐶 𝑐 𝑥𝑠,𝑚,𝜏 ′ + 𝜏 ≥ 𝑦𝑠,𝜏 − (1 − 𝑧𝑠,𝑚 )M − ( 𝑐 1 − 𝑟 , 𝑠,𝑐 )M (17) 3 2 3 0.56 3 3 2 1.50 where ′ ′ + denotes the time units after time . 3 3 3 0.67 𝜏 𝜏 𝜏 𝜏 (8) Merged operations can not be active earlier than they start. 5 2 2 6.90 ′ For each merged operation 𝑠 ∈ 𝑆 , for each 𝜏 ∈ 𝑇 and 𝜏 ∈ 𝑇 5 2 3 2.55 ′ such that 𝜏 < 𝜏 : 5 3 2 6.36 𝑥 ′ 𝑦 ≤ (18) 5 3 3 3.09 𝑠,𝑚,𝜏 𝑠,𝜏 7 2 2 16.81 (9) Machines are not assigned merged operations during the periods of inactivity. For each time unit 7 2 3 7.54 𝜏 ∈ 𝑇 \ 𝐴 : ∑︁ 7 3 2 20.33 𝑥 𝑠,𝑚,𝜏 = 0 (19) 7 3 3 10.70 𝑠 𝑆 ,𝑚 𝑀 ∈ ∈ 10 2 2 124.92 (10) For each operation 𝑜 ∈ 𝑂 only one merged operation 𝑠 ∈ 𝑆 10 2 3 40.62 such that 𝑜 ∈ 𝑠 can be active during the time horizon. For 10 3 2 156.67 each 𝑜 ∈ 𝑂: 10 3 3 47.94 ∑︁ 𝑟𝑠 = 1 (20) Table 1: Average runtimes (in seconds) for OSSP-OB in- 𝑠 𝑆 ,𝑜 𝑠 ∈ ∈ stances. 3.4 Objective function Minimize the overall tardiness of merged operations with respect to internal deadline of the corresponding jobs: Future research will therefore focus on approximation meth- ods. A potential approach to consider is separating the OSSP-OB Minimize © ª 𝑘 𝑠,𝑚, 𝑗 ,𝜏 ­ ® (21) scheduling of the merged operations, combining exact methods environment into two subproblems, merging of operations and ∑︁ ∑︁ ∑︁ ∑︁ 𝑠 𝑆 𝑚 𝑀 𝑗 𝐽 𝑑 𝜏 ∈ ∈ ∈ ∈ 𝑇 ,𝜏 ≥ 𝑡 𝐽 (𝑗 ) « ¬ for the two parts separately. Another approach is to design dedi- 4 cated (meta-)heuristics specific to the merging structure. Such RESULTS approaches are expected to extend the range of solvable instances To evaluate our approach, we generated a new benchmark set and provide practical solutions for larger problem sizes. for the OSSP-OB following the ideas of E.Taillard[6]. Instances are defined by the number of jobs, machines, and categories. We ACKNOWLEDGEMENTS considered jobs of sizes 3, 5, 7, and 10, and for each case the num- The research was supported by the BioLOG project: Balázs Dávid ber of machines and categories was set to either 2 or 3. All other and Miklós Krész are grateful for the support of National Center parameters, such as processing times, time horizon, and number of Science (NCN) through grant DEC-2020/39/I/HS4/03533, the of operations per job, were kept constant. The construction pro- cedure is flexible and can be extended to introduce randomness Slovenian Research and Innovation Agency (ARIS) through grant N1-0223 and the Austrian Science Fund (FWF) through grant or additional parameters if needed. Each valid merge corresponds to a subset of operations of the I 5443-N. This work was partially supported by the Slovenian Research Agency, research program P1-0404 and by the research same category, meaning that the set of possible merges is a subset of the power set 𝑂). The size of this set is strongly influenced P ( program CogniCom (0013103) at the University of Primorska. Balázs Dávid is grateful for the support of ARIS through grant by the number of available category capacities: smaller values J1-50000, and gratefully acknowledges the Slovenian Research lead to larger collections of potential merges. This effect directly impacts solver memory usage and thus scalability. and Innovation Agency (ARIS) and the Ministry of the Economy, Tourism and Sport (MGTŠ) for the grant V4-2512. Computational experiments were carried out using the Gurobi 11.0.0 solver on an AMD Ryzen 7 5800𝐻 3.2 GHz CPU with 16 REFERENCES GB RAM, with a time limit of 60 minutes per instance. Table 1 demonstrates the maximum runtime observed for each instance [1] Danyu Bai, Zhi-Hai Zhang, and Qiang Zhang. 2016. Flexible open shop scheduling problem to minimize makespan. Computers & operations research, configuration. All instances of this set were solved optimally. 67, 207–215. Overall, the model is able to solve instances up to 10 jobs [2] John W Fowler and Lars Mönch. 2022. A survey of scheduling with parallel batch (p-batch) processing. European journal of operational research, 298, 1, in a reasonable time, but the growth of the merged operation 1–24. space remains the main computational bottleneck. For example, [3] Wieslaw Kubiak. 2022. Book of Open Shop Scheduling. Springer. increasing the number of jobs to 15 leads to out-of-memory [4] B Naderi, SMT Fatemi Ghomi, Majid Aminnayeri, and Mostafa Zandieh. 2011. Scheduling open shops with parallel machines to minimize total completion errors. time. Journal of Computational and Applied Mathematics, 235, 5, 1275–1287. [5] Chris N Potts and Mikhail Y Kovalyov. 2000. Scheduling with batching: a 5 review. European journal of operational research, 120, 2, 228–249. CONCLUSION [6] Eric Taillard. 1993. Benchmarks for basic scheduling problems. European In this work, we introduced a complete MILP formulation for journal of operational research, 64, 2, 278–285. OSSP-OB. Our experiments show that the model can solve small stage batching and scheduling. In [7] Tanya Y Tang and J Christopher Beck. 2020. CP and hybrid models for two- Integration of Constraint Programming, instances to optimality within a reasonable time, but quickly Artificial Intelligence, and Operations Research: 17th International Conference, becomes impractical as the number of jobs and possible merges CPAIOR 2020, Vienna, Austria, September 21–24, 2020, Proceedings 17. Springer, 431–446. increases. Rapid growth of the set of potential merged operations leads to excessive memory usage and limits scalability. 10 Flexibility vs. efficiency: a study in sawmill scheduling Csaba Kebelei Máté Hegyháti Eötvös Loránd University University of Sopron Faculty of Informatics Sopron, Hungary Budapest, Hungary University of Pannonia kebeleicsaba@student.elte.hu Veszprém, Hungary Abstract scheduling problem. The base problem was introduced by Zanjani Sawmills are an important stage in the primary wood industry, et al. [7] and Maturana et al. [9] for the stochastic and deterministic producing timber and other side products from harvested logs. In- cases, respectively. Subsequent works extended the approach in creasing efficiency in a sawmill is not only an economic desire, but various directions, e.g., addressing uncertainties [2, 11], integrating it also propagates to the competitiveness of using more renewable cutting pattern selections [3, 10], or minimizing waste [4]. resources in dependent industries such as construction, packaging, furniture. Although the road from a tree log to a shipped product involves many stages, the key processing step is the sawing per- formed by high-value sawing machines. As the operation of these machines plays a key role in overall efficiency, several research papers have addressed their scheduling. Naturally, allowing more freedom in the production plan may result in better financial results in exchange for increased computational needs of the optimizer. However, more complex schedules may also pose an additional bur- den in their real-life execution. This paper presents several variants of a model formerly proposed by the authors and investigates this trade-off relationship empirically. Keywords sawmill, scheduling, MILP 1 Introduction and literature The increasing utilization of renewable resources is a major direc- tive of many nations and companies around the world. Wood is a natural resource that not only serves as a great carbon sequestration tool, but it is a suitable raw material for many industries, including construction and interior design, paper, packaging, and the boat industry. After logging, sawmills are generally the first stage in timber processing regardless of the end-use industry. Their main task is to turn unprocessed logs from the forest into well-cut lumber. While this is a multi-stage process, the main step is the sawing itself that can be done by two of the main machine types: band saws and frame saws. The efficient operation of these machines have a significant impact on the whole plant, thus, several works have addressed this issue. Efficiency is tackled on both the operational and the planning level by designing cutting patterns and schedul- ing, respectively [6]. A tree log may be cut in different patterns yielding different quantities of different products, as shown by in Figure 1: Cutting patterns for test cases in Section 5 Figure 1. Designing patterns based on long-term demands for prod- ucts, statistics about defects, etc. is a well-researched topic in the In this paper, a previously proposed model by the authors [8] is literature. Moreover, scanning tools available on modern sawing further investigated, that addressed issues prevalent for small-scale machines can provide real-time information about logs to make sawmills: workforce availability and operation differences between such decisions adaptive [5]. On the planning level, selecting the sawing technologies. Compared to band saws, the changeover time cutting patterns and the corresponding log quantities for each shift for switching cutting patterns on a frame saw is non-negligible, thus while considering orders, available logs, storage capacity, etc. is a the same formulation is not applicable in short-term scheduling. MATCOS-25, Koper, Slovenia The proposed model introduced a significantly different formulation 2025. for the behavior of the frame saw, that only allows one change per 11 MATCOS-25, October 09–10, 2025, Koper, Slovenia Kebelei and Hegyháti shift following industry practice. In later Sections this base model 3 Model with workforce reallocation: C’–EK will be referred to as C–EK, where EK (expert knowledge) indicates The event, when worker reallocation is also possible in addition at most one change in a shift, that may be timed continuously (C) to pattern changes on the frame saw, will be referred to as the at any time. changeover. As this model is based on C–EK, constraints (1)–(5), The aim of this work is to investigate, whether allowing more (7)–(8), and (10) from [8] are copied verbatim, expressing the objec- cutting pattern changes on a frame saw would yield significantly tive, storage balances, daily production quantities, inventory limits, better results due to a broader solution space, and/or does it in- frame saw production bounds before the changeover, and frame crease the computational costs comparably. While allowing such saw activation bounds. flexibility in a mathematical model may be simple, one has to keep 𝐹 𝐵 𝐹 ,− 𝐵,− 𝐹 ,+ 𝐵,+ Binary variables 𝑤 𝑤 𝑑 and 𝑤 𝑑 are split to 𝑑 , 𝑤 𝑤 𝑑 and 𝑑 , 𝑤 𝑑, in mind, that the execution of overly complicated schedules in an indicating whether the machines are active before or after the non-automated small-scale plant is not realistic, thus a set of models changeover. Duplicated versions of constraints (15)–(16) with these will also be introduced, where cutting pattern changes may only variables are also copied to express human resource needs. Su- happen at predefined discrete (D) time points, as detailed in Section perscripts + and − will indicate such division similarly in other 4. Another decision freedom, whose effect will be investigated in variables, and ± will be used to indicate both cases. Section 5 is the possible reallocation of the workforce from one The band saw time capacity constraint is reformulated as: machine to another during a shift, as discussed in Section 3. ∑︁ 𝐵, 𝐵 𝐵− 𝐵,+ · ≤ + ∀ ∈ D (1) 𝑆𝑇 𝜏 𝑞 𝜏 𝑑 2 𝑝 𝑑 ,𝑝 𝑑 𝑑 Problem definition 𝑝 ∈ P The objective is the same as in [8]: minimizing the under-production 𝐵,± Where variables 𝜏 ∈ [0, 𝐻 ] represent the available band saw cost in a small-scale sawmill equipped with one frame saw and one 𝑑 cutting time, linked to changeover timing by the following con- band saw. The sawmill may operate in multiple shifts, however, straints: from the modeling point of view, only the total number of shifts is relevant. Thus, without the loss of generality, it is assumed, that 𝐵,± 𝐵,± 𝜏 ≤ 𝐻 · 𝑤 ∀𝑑 ∈ D, ± ∈ {−, +} (2) 𝑑 each day has a single shift (of length 𝑑 𝐻 ), and the two terms are used interchangeably. For each day (𝑑 ∈ D), the number of specialists 𝐵,− ( 𝑆 𝑃 𝑑 𝜏 𝑡 𝑑 ≤ ∀ ∈ D 𝐻 𝑅 𝐻 𝑅 ) and additional workers ( ) are given, along with the 𝑑 𝑑 𝐴𝑊 𝑑 (3) 𝑅𝑅 requirements to operate each machine ( 𝑆 𝑃 ,𝐹 𝑆 𝑃 ,𝐵 𝑅𝑅 , 𝑅𝑅, 𝐴𝑊 ,𝐹 𝑅𝑅 , 𝐵,+ 𝜏 ≤ 𝐻 − 𝑡 𝑑 𝑑 ∀ ∈ D (4) 𝑑 𝐴𝑊 ,𝐵 ). Resource related data is also given, i.e., the quantity ( 𝐼 𝑙 ) and volume (𝑉 𝐵, 𝑙 ) of logs of different sizes ( 𝑙 ∈ L ) all available at± Constraint (2) activates 𝜏 𝑑 only when the band saw is active the start of the planning horizon. For each log size several cutting in the corresponding part of the shift, while constraints (3) and (4) patterns ( P 𝑙 ⊆ P ) may be available with known yields ( 𝑌 𝑡 ,𝑝 ) for bound them by 𝑡 𝑑 and 𝐻 − 𝑡 𝑑 , respectively. lumber products ( 𝑡 ∈ T ). For each shift the production planner may For the frame saw, the following constraint limits the production decide which machines to operate and how many logs with which after the changeover: cutting patterns are processed. 𝑆𝑇 𝑞 𝐻 𝑡 𝐶𝑇 𝑧 𝐻 𝑤 ≤ ( − ) − · + · 1 − nitions will be addressed by their corresponding models: 𝑝 𝑑 𝑑 ,𝑝 𝑑 𝑑 ∈ P · By allowing different flexibilities, several different problem defi- ∑︁ 𝐹 𝐹 , + 𝐹 𝐹 ,𝐶𝑇 𝐹 , + 𝑝 (5) C–EK Original problem from [8]: the frame saw may change ∀𝑑 ∈ D cutting pattern once per shift, but a saw is either operating 𝐹 ,𝐶𝑇 through the whole shift or not. Where 𝑧 ∈ { 𝑑0, 1} indicates whether changeover time must C ′ 𝐹 ,𝐶𝑇 𝑧 –EK be deducted after the changeover. Constraint (6) sets Same as C–EK , but the frame saw may also be turned to one, 𝑑 on/off, and the workforce reallocated to the band saw instead if the frame saw is active through the whole shift and the pattern of changing the pattern, once per shift. is changed. D[𝐾] The shift is subdivided into 𝑘 segments of equal length. − 𝐹 ,𝐶𝑇 𝐹 , 𝐹 , + 𝐹 , + 𝐹 , − Pattern changes and workforce reallocations are allowed 𝑧 ≥ 𝑤 + 𝑤 + 𝑠 − 𝑠 − 2 ∀𝑑 ∈ D, 𝑝 ∈ P (6) 𝑑 𝑑 𝑑 𝑑 ,𝑝 𝑑 ,𝑝 only at the end of these intervals. D[𝐾]–EK Same as D[𝐾], but the frame saw is allowed to change Constraint (7) ensures that only an active frame saw can have cutting patterns at most once a day. an active cutting pattern, which is preserved by constraints (8) and If (9) for the next day if the saw remains active. 𝑂 𝑃𝑇 ( 𝑀 ) denotes the optimal (minimal) solution for the prob- lem/model 𝑀, the following inequalities will hold naturally: ∑︁ 𝐹 ,± 𝐹 ,± 𝑠 𝑤 𝑑 , , = ∀ ∈ D ± ∈ {− +} (7) • ′ 𝑑 ,𝑝 𝑑 ( C –EK ) ≤ ( C–EK ) 𝑂 𝑃𝑇 𝑂 𝑃𝑇 ∈ P 𝑝 • 𝑂 𝑃𝑇 ′ ( C–EK) ≤ 𝑂 𝑃𝑇 (D[𝑋 ]–EK) + 𝐹 , 𝐹 , − 𝐹 , + 𝐹 , − ≥ 𝑠 𝑠 − · − 𝑤 𝑑 , 𝑝 + 𝑑 ,𝑝 𝑑 ,𝑝 1 2 𝑤 − ∀ ∈ D ∈ P • 𝑂 𝑃𝑇 ( D1 ) = 𝑂 𝑃𝑇 ( D1–EK ) 1 𝑑 𝑑 (8) + 1 • 𝑂 𝑃𝑇 (D[𝑋 ]) ≤ 𝑂 𝑃𝑇 (D[𝑋 ]–EK) + 𝐹 , 𝐹 , − 𝐹 , + 𝐹 , − ≤ • 𝑠 𝑂 𝑃𝑇 ( D[ 𝑠 𝑌 + ] ) ≤ 𝑂 𝑃𝑇 ( D[ 𝑋 ] ) 1 · 2 − if 𝑤 𝑋 − | 𝑤 𝑌 ∀𝑑 ∈ D, 𝑝 ∈ P (9) 𝑑 ,𝑝 𝑑 + 1 ,𝑝 𝑑 𝑑 + 1 • 𝑂 𝑃𝑇 (D[𝑌 ]–EK) ≤ 𝑂 𝑃𝑇 (D[𝑋 ]–EK) if 𝑋 | 𝑌 12 Flexibility vs. efficiency: a study in sawmill scheduling MATCOS-25, October 09–10, 2025, Koper, Slovenia 4 Discrete time-slot models: D[𝐾 ], D[𝐾 ]–EK 𝑆𝑇 𝑞 𝐻 𝐶𝑇 𝑠 𝐻 𝑠 𝑝 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 ,𝑝 Similarly to the previous modification of the original model, some 𝐹 𝐹 𝐾 𝐹 𝑐,𝐹 𝐾 𝐹 · ≤ − · + · constraints from [8] remain unmodified, expressing the objective, + 𝜎 𝜎 𝑑 ,𝑘 − 1 ,𝑝 −𝑑 ,𝑘 ,𝑝 (18) storage balances: (1)–(3). Moreover, some variables are now defined for each segment of a shift, indexed by the set ∀𝑑 ∈ D, 𝑘 ∈ K \ {1} , 𝑝 ∈ P K = { 1 , 2 , . . . , 𝐾 } . For example, the binary variables 𝐹 𝐵 indicate machine Constraint (16) limits the carryover from segment to one log’s 𝑤 , 𝑤 𝑘 𝑑 ,𝑘 𝑑 ,𝑘 usage, which are used in segment-wise copies of (15)–(16) of [8] to sawing time 𝐹 𝑆𝑇 𝑝 and activates it only if the next segment continues express human-resource requirements. with the same pattern, while the production time bounds follow 𝑤 𝐵 𝐹 𝐶𝑇 from Eqs. (17) and (18), accounting for changeover time is also used in the band saw time capacity constraint, where and 𝑑 ,𝑘 𝐾 the inflow/outflow of spare time. denotes the length of a segment: 𝐻 𝐻 𝐾 = / To obtain D[𝐾]–EK from D[𝐾], Constraint (19) needs to be added, ∑︁ ∑︁ which enforces at most one cutting pattern change by the frame 𝐵 𝐵 𝐾 𝐵 · ≤ · ∀ ∈ D (10) 𝑆𝑇 𝑞 𝐻 𝑤 𝑑 𝑝 𝑑 ,𝑝 𝑑 ,𝑘 saw per day: 𝑝 𝑘 ∈ P ∈ K Another such variable is 𝐹 ∑︁ ∑︁ 𝑞 ∈ 𝑑 ,𝑘 ,𝑝 Z ≥ 0 indicating the frame saw 𝑐,𝐹 𝑠 ≤ ∀ ∈ D 1 𝑑 ,𝑘 ,𝑝 𝑑 (19) production, used in constraints (11) and (12) to calculate the daily 𝑘∈ K 𝑝∈ P production quantity and limit the inventory: ! 5 Empirical results ∑︁ ∑︁ 𝑦 𝑌 𝑞 𝑡 ,𝑝 𝑑 ,𝑡 = 𝐵 𝐹 · · + ∀ ∈ D ∈ T (11) To evaluate the new approaches, the randomly generated test set 𝑉 𝑑 𝑞 , 𝑡 𝑙 𝑑 ,𝑝 𝑝 𝑑 ,𝑘 ,𝑝 𝑝 𝑘 ∈ P ∈ K from our previous study [8] is used, where cutting pattern yields ! were precomputed with Pitago Optimizers [1] and technical pa- ∑︁ ∑︁ ∑︁ 𝐵 𝐹 + ≤ ∀ ∈ L (12) rameters reflect industry practice and recommendations by experts. 𝑞 𝐼 𝑙 𝑞 𝑑 ,𝑝 𝑑 ,𝑘 ,𝑝 𝑙 𝑑 𝑝 𝑘 ∈ D ∈ P∈ K Figure 1 shows the five patterns defined for three log diameter 𝑙 Active consecutive segments on the frame saw should be merged classes in all of the test cases. together if the pattern is unchanged. These conditions are expressed For practical reasons in execution, the values 1, 2, 4, and 8 were by binary variable 𝐹 𝑠 which is 1 iff pattern 𝑝 considered for 𝐾, and the EK is active in segment variant for 𝐾 = 1 is omitted, as it is 𝑑 ,𝑘 ,𝑝 𝑑 , 𝑘 equivalent to the non-EK case. and the one preceding it. Another binary indicator variable, 𝑐,𝐹 All problems in the dataset were solved with all of the models is 1 iff is active in segment but not in the preceding one. 𝑠 𝑝 𝑘 𝑑 ,𝑘 ,𝑝 by Gurobi Optimizer 12.0.1 on a computer with an Apple M2 CPU This logic is enforced by the following constraints: and 16 GB of RAM available, with a time limit of 1500 seconds. ∑︁ 𝑐,𝐹 𝐹 𝐹 Table 1 shows the aggregated results from the 50 cases with 𝑠 𝑠 𝑤 𝑑 , 𝑘 + = ∀ ∈ D ∈ K (13) 4-week planning horizon. 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 𝑝 ∈ P 3 The left block shows objective value distributions (m) as box 𝐹 𝐹 𝑐,𝐹 𝐹 ≤ + + 1 − plots. The median relative deviation from the C–EK reference is also 𝑠 𝑠 𝑠 𝑤 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 ,𝑝 𝑑 ,𝑘 − 1 − 1−1 (14) reported in the column labeled ˜ ΔC–EK (%). On the right, CPU times ∀𝑑 ∈ D, 𝑘 ∈ K \ {1}, 𝑝 ∈ P (s) are shown as box plots on a base-10 logarithmic scale. 𝐹 𝐹 𝑐,𝐹 𝐹 ≤ + + − The results match the theory: 1 𝑠 𝑠 𝑠 𝑤 1 − 1 − 1 1 𝑑 , ,𝑝 𝑑 ,𝐾 ,𝑝 𝑑 ,𝐾 ,𝑝 𝑑 ,𝐾 − (15) ∀ C –EK • ′ Adding on/off reallocation in 𝑑 , 𝑝 ∈ D \ { 1 } increases quality ∈ P • Discretizing the changeover timing degrades it when only Constraint (13) ensures that when the frame saw is operated in one cutting pattern change is allowed per shift (EK models), segment 𝑘, exactly one pattern is selected (either a changeover to while smaller slot lengths mitigate the loss. 𝑝 𝑝 or continuation with), and when it is idle, neither indicator is • Imposing the “at most one change” rule on the frame saw active. Constraint (14) allows pattern continuation only if 𝑝 was (–EK) has a negative impact on solution quality. active in the previous segment, and the frame saw is active. The same condition is set across days is by Constraint (15). It can be observed, however, that the differences in objective The merging of active segments with the same pattern is modeled values across models are small: ˜ ΔC–EK ranges from −0.1072% for C′–EK (best) to +0 8536% for (worst), and difference between by constraints (16), (17), and (18), introducing a spare time variable . D1 𝜎 , 𝐻 𝑘 0 ] that records unused time in segment 05 percentage points. for pattern 𝑑 ,𝑘 ,𝑝 ∈ [ 𝐾 𝐾 𝐾 . D[ ] and D[ ]–EK is less than 0 𝑝 A key takeaway is that even D8 does not outperform C–EK. For and can be carried to the next segment. these test cases, continuously timing a single change is more valu- 𝐹 𝐹 able than permitting multiple changes at fixed time-slot boundaries. (16) 𝜎 𝑆𝑇 𝑠 𝑑 , 𝑘 𝐾 , 𝑝 𝑑 ,𝑘 ,𝑝 𝑝 𝑑 ,𝑘 + ≤ · ∀ ∈ D ∈ K \ { } ∈ P 1,𝑝 𝑘 Even so, the loss from discretization is small: as increases, the 𝐹 . 𝐹 𝐾 𝐹 𝑐,𝐹 𝐾 𝐹 median gap shrinks from + 085% (D1) to about +0.0058% for D8–EK 𝑆𝑇 𝑞 𝐻 𝐶𝑇 𝑠 𝐻 𝑠 𝜎 𝑝 𝑑 ,1,𝑝 𝑑 ,1 ,𝑝 ,𝑝 𝑑 , 1 ,𝑝 𝑑 , 1 (17) or · ≤ − · + · − +0.0024% for D8. ∀𝑑 ∈ D, 𝑝 ∈ P Regarding runtimes, increasing 𝐾 for the discrete models in- creases CPU time, as expected, yet the 𝐾 = 8 variants still run faster 13 MATCOS-25, October 09–10, 2025, Koper, Slovenia Kebelei and Hegyháti Table 1: Results on randomly generated instances with a 4-week planning horizon across model variants. Objective value (m3) CPU time (s) Model 950 1025 1100 1175 1250 Δ ˜ −2 −0.625 0 75 2 125 3 5 . C—EK (%) 10 10 10 . 10 . 10 Avg. C–EK +0.0000 151.36 C′–EK -0.1072 125.98 D1 +0.8536 0.24 D2–EK +0.2908 1.37 D2 +0.2440 1.52 D4–EK +0.1065 5.00 D4 +0.0760 4.21 D8–EK +0.0058 67.76 D8 +0.0024 82.82 on average than the continuous baselines. Notably, D8 is within with similar quality and often lower runtime. For the investigated ≈ ′ 0 11 percentage points of C–EK in objective value yet is faster on test cases, even completely forbidding changes within shifts resulted . average. The –EK restriction leaves quality essentially unchanged in less than 1 percent quality reduction. These findings support and tends to reduce runtime. the discrete-slot approach as a practical planning tool for small, Time-limit hits were infrequent (Table 2). low-automation sawmills. Future work will expand the set of test instances, include an industrial case study to validate and calibrate Table 2: Time-limit hits (counts). the models on operational data, and, where appropriate, incorporate uncertainty in key inputs. Model 3-week 4-week 5-week 6-week References C–EK 3 2 5 4 [1] Pitago Optimizers. Retrieved August 24, 2025 from https://pitago.eu/ C′ [2] Pamela P Alvarez and Jorge R Vera. 2014. Application of robust optimization to –EK 0 1 2 3 the sawmill planning problem. Annals of Operations Research 219 (2014), 457–475. D8–EK 0 1 1 0 doi:10.1007/s10479-011-1002-4 D8 0 1 1 1 [3] Diego Broz, Nicolás Vanzetti, Gabriela Corsano, and Jorge M. Montagna. 2019. Goal programming application for the decision support in the daily production planning of sawmills. Forest Policy and Economics 102 (2019), 29–40. doi:10.1016/ These counts suggest that the discrete-slot models remain com- j.forpol.2019.02.004 [4] Giacomo Da Col, Philipp Fleiss, Alice Tarzariol, Erich C. Teppan, and Elena putationally stable across longer planning horizons. Wiegelmann. 2025. A Declarative Approach to Tackle Sawmill Production Sched- Overall, the discrete time-slot-based models offer a tunable trade- uling with Answer Set Programming. In Computational Science and Computational off: coarse slotting is extremely fast but less accurate, whereas 𝐾 = 8 madi, Soheyla Amirian, and Farzan Shenavarmasouleh (Eds.). Springer Nature Intelligence, Hamid R. Arabnia, Leonidas Deligiannidis, Farid Ghareh Moham- achieves near-continuous quality at a substantially lower average Switzerland, Cham, 313–323. doi:10.1007/978-3-031-90341-0_23 runtime than the continuous baselines. [5] Kamran Forghani, Mats Carlsson, Pierre Flener, Magnus Fredriksson, Justin Pearson, and Di Yuan. 2024. Maximizing value yield in wood industry through flexible sawing and product grading based on wane and log shape. Computers and 6 Concluding remarks Electronics in Agriculture 216 (2024), 108513. doi:10.1016/j.compag.2023.108513 [6] Seyed Mohsen Hosseini and Angelika Peer. 2022. Wood Products Manufacturing We revisited a MILP scheduling model for small-scale sawmills and Optimization: A Survey. IEEE Access 10 (2022), 121653–121683. doi:10.1109/ examined three design choices: (i) allowing workforce reallocation, ACCESS.2022.3223053 (ii) restricting changeovers to discrete within-shift time-slot bound- [7] Masoumeh Kazemi Zanjani, Daoud Ait-Kadi, and Mustapha Nourelfath. 2010. Robust production planning in a manufacturing environment with random yield: aries, and (iii) allowing multiple changes per shift. Computational A case in sawmill production planning. European Journal of Operational Research tests on a large synthetic instance set show that objective differ- 201, 3 (2010), 882–891. doi:10.1016/j.ejor.2009.03.041 ences across variants are modest. Allowing on/off operation with oriented model. Acta Technica Jaurinensis [8] Csaba Kebelei and Mate Hegyhati. 2024. Sawmill scheduling: an application- 17, 3 (2024), 104–110. doi:10.14513/ reallocation provides a consistent, albeit small, improvement in so- actatechjaur.00743 lution quality, whereas discretizing the changeover into time slots [9] Sergio Maturana, Enzo Pizani, and Jorge Vera. 2010. Scheduling production for a sawmill: A comparison of a mathematical model versus a heuristic. Computers & introduces a controllable loss that diminishes as the slot granularity Industrial Engineering 59, 4 (2010), 667–674. doi:10.1016/j.cie.2010.07.016 is refined. Overall, the discrete formulations offer a tunable trade- [10] Nicolás Vanzetti, Diego Broz, Gabriela Corsano, and Jorge M. Montagna. 2018. off between accuracy and speed. In our setting, even coarse slots An optimization approach for multiperiod production planning in a sawmill. Forest Policy and Economics 97 (2018), 1–8. doi:10.1016/j.forpol.2018.09.001 perform well, while a moderate refinement (e.g., hourly segments) [11] Mauricio Varas, Sergio Maturana, Rodrigo Pascual, Ignacio Vargas, and Jorge achieves near-continuous quality at substantially lower average Vera. 2014. Scheduling production for a sawmill: A robust optimization approach. runtimes. In practice, multiple within-shift changes bring marginal ijpe.2013.11.028 International Journal of Production Economics 150 (2014), 37–51. doi:10.1016/j. gains and complicate execution, whereas –EK yields simpler plans 14 Implementation of a Vehicle and Driver Scheduling Model: a Case Study Viktor Árgilán József Békési University of Szeged, University of Szeged, Institute of Informatics, Juhász Gyula Faculty of Education, Department of Foundations of Computer Science Department of Applied Informatics Szeged, Hungary Szeged, Hungary bekesi@inf.u-szeged.hu viktor.sandor.argilan@szte.hu Gábor Galambos Imre Papp University of Szeged, University of Szeged, Juhász Gyula Faculty of Education, Juhász Gyula Faculty of Education, Department of Applied Informatics Department of Applied Informatics Szeged, Hungary Szeged, Hungary GalambosGabor@szte.hu Abstract The Generate and Select (GaS) method is the most well- During combined vehicle and driver scheduling, we have to known technology for the driver scheduling part. In the plan the daily work of vehicles and their drivers so that they initial phase, a substantial number of standard shifts are perform a set of tasks at the lowest possible cost. The tasks generated. In the subsequent selection phase, a subset of are defined by time intervals, and the vehicles are located these regular shifts is selected to minimize cost and optimize in different depots. In recent decades, several mathematical coverage of trips. It is noteworthy that the execution of both models have been defined, with which we can create reg- phases requires substantial computational resources. The ular schedules and search for optimal solutions based on extent of this computational demand is contingent upon different objective functions. However, in practical prob- the number of trips and the intricacy of the operational lems, there are many requirements that cannot be handled guidelines. The selection phase can be modeled as a set easily and usually have a significant computational demand. covering or set partitioning problem. The results of our research and development project are In 2005, Huisman et al. [9] extended the former combined presented through a case study based on real experiments models and algorithms of the single-depot case [5, 6] to the carried out at the Budapest Transport Corporation. multi-depot version. This was the first general mathematical formulation of the combined multi-depot problem. Later Keywords many authors investigated this version of the problem (see, for example, [8], [13], [15], [16]). Optimization, Vehicle and driver scheduling problem, Pub- In practice, however, it turns out that there are many lic transport company-specific details and constraints that cannot be uniformly addressed by general systems, but which are 1 Introduction important for the transport companies. As an example, if a Operating costs are usually a significant item in the bud- transport company also uses alternative fuel vehicles in its get of public transport providers. The main components fleet, their scheduling must take into account the number of these costs are vehicle fleet acquisition costs, fuel and of kilometers per refueling, known as the radius, which maintenance costs, and driver wages. With the help of var- can be much lower than the mileage of a conventional fuel ious decision support systems, comprehensive solutions vehicle. Such cases have been examined for example in [1] for both the vehicle and the driver have been developed and [11]. in recent decades to solve the optimization task. In public In the paper [3] Békési and Nagy presented how the transport, vehicle and driver scheduling can be very com- methods used in the above mentioned papers were adapted plex. In theory, we generally look for a global optimum that to develop a decision support system for the Budapest Trans- minimizes both vehicle-related costs and driver scheduling port Corporation. The aim of this project was to automat- costs. These two types of costs affect each other, so it’s ically calculate optimal or approximately optimal vehicle usually best to handle the tasks together [2]. and driver schedules for a given list of trips based on the If we want to find the optimal solution, combined vehicle master data and the company specific requirements and and driver scheduling mathematical optimization models parameters in compliance with labor regulations. can be used. There are several such methods in the literature. This paper overviews how the complete integration was The vehicle scheduling problem is usually formulated as implemented and what kind of specific developments were a multicommodity network flow problem ([4], [10], [12]). necessary to take into account all the practical requirements The optimal schedule can be calculated as the solution of of the company. an integer programming problem. Other models are also known, for example the problem can be formulated as a set 2 The Automatic Solving Process partitioning problem (see e.g. [14], [7]). 15 ments, input data, and settings based on [3]. All the data We summarize the system’s key characteristics, require- MATCOS-25, Koper, Slovenia 2025. https://doi.org/10.1145/nnnnnnn.nnnnnnn required for a computation is kept in packages so that it MATCOS-25, October 9–10,2025, Koper, Slovenia Viktor Árgilán, József Békési, Gábor Galambos, and Imre Papp can automatically solve a particular problem. Trips from a - The generation is complete if every thread is termi- single line or a collection of lines are typically included in nated. a package. The following details are included in each input package: 3.2 Smart node contraction • line data, According to [3], the greedy strategy is the fundamental • end stations, depots and their parameters, method of node-contraction. This method allows us to elim- • trips with details about the time and place, inate a large number of edges from the graph. As a result, • number of vehicles along with details about their several opportunities for breaks and driver changes are also type and availability, eliminated. It can occasionally result in infeasibility. We • the limits of labor laws and break rules, created an intelligent node-contraction algorithm after re- • parking capacities of stations, parking lots, and de- alizing this issue. Compared to the greedy approach, this pots provided at 5-minute intervals for each type of algorithm retains more break and driver change possibili- vehicle ties based on its parameter values. The pseudocode of the • driver change possibilities, break and detour permis- algorithm is the following. sions The following conditions must be met by the problem’s solution: Algorithm 1 Smart Trip Grouper • 1: procedure groupTripsSmart( every trip must be covered by a single vehicle and𝑛:Integer, 𝑆 : Set of driver schedule, Trips) • 2: for all the number of vehicles specified in the package is 𝑇 ∈ 𝑆 do the maximum number that can be used, 3: Next(𝑇 ) ← The closest compatible trip to 𝑇 • schedules for drivers and vehicles must be consistent, 4: for all 𝑇 ∈ 𝑆 do • parking regulations and fuel consumption must be 5: if Merged(𝑇 ) = false then followed, 6: actTrip ← 𝑇 • between two trips, the necessary technological and 7: MList ← ∅ compensatory times must be maintained, 8: 𝑖 ← 1 • driver change regulations must be followed. 9: while 𝑖 ≤ 𝑛 do There are three phases in our process. In the first phase 10: if 𝑖 > 1 and actTrip.time - prevTrip.time a directed graph is produced after the relevant input data ≥ MAXT_NOPROTECT then 11: if actTrip.hasInBreakArc or pre- and parameters have been read. We consider a number of "side-conditions" when creating the graph, including vTrip.hasOutBreakArc then 12: Exit while location data, vehicle types, labor laws, technological and compensatory times, and other package elements. In the 13: if actTrip.hasInDriverChArc or pre- vTrip.hasOutDriverChArc then Here, only schedules that comply with all labor laws are 14: Exit while ← second phase all regular driver schedules are generated. 15: Merged(actTrip) true mathematical model. The details of the model are available 16: MList ← MList ∪ actTrip ← accepted. The third phase involves building and solving a in [3]. If the solution is successful, an output package is 17: prevTrip actTrip used to send the results back to the company’s information 18: 𝑖 ← 𝑖 + 1 ≠ 19: if Next(actTrip) null then system after being read from the solver. 20: actTrip ← Next(actTrip) 3 21: else Implementation details 22: Exit for 3.1 Parallel schedule generation 23: Add MList to the output We employ the depth-first search strategy to generate the regular shifts, beginning with the special departure depot vertices of the graph. We implemented the shift genera- Similar to the greedy strategy, as the initial step of the tion as a parallel algorithm because it can be quite time- algorithm, for each trip vertex in the graph, we search for consuming. At this point, the paths of the graph that begin the closest trip vertex to it. As a result, we get chains of at the departure depot vertex and conclude at the arrival subsequent trip vertices here as well. After that we take the depot vertex reflect the shifts. As a first stage for the paral- trip vertex 𝑣 that have not been included in a group before. lel generation, we create and save in a list every potential Starting from 𝑣 , we move forward on the chain at most 𝑛 −1 prefix of the paths from the departure depot vertex to a times and decide whether we include the actual vertex in specified short depth (for example, 4). the current group or not. If the time difference between the The following is a summary of the procedure. time of the previous and the current trip vertex is less than - Each thread chooses an unprocessed path-prefix from or equal to the value of the parameter MAXT_NOPROTECT, the list and uses it to create every potential regular then we proceed as for the greedy strategy. Otherwise, if shift in the main portion of the generation. the vertex before the actual vertex has an outgoing break - Following the completion of the operation, the thread arc, or the actual vertex has an incoming break arc, then shifts, and saves the generated shifts. 16 merging with vertex v. We do the same with the driver chooses a fresh unprocessed prefix, generates its we stop and the actual vertex is no longer included in the - The thread ends if no shifts remain unprocessed. change arcs. Implementation of a Vehicle and Driver Scheduling Model: a Case Study MATCOS-25, October 9–10,2025, Koper, Slovenia 3.3 Parking and vehicle number 𝑑 𝑥 𝑒 Boolean variable indicating whether the constraints for several types of arc 𝑒 from depot 𝑑 is included in the solu- vehicles tion or not. 𝑑 Boolean variable indicating whether the 𝑦 𝑠 The parking capacities for various vehicle types are also schedule 𝑠 ∈ 𝑆𝑑 is included in the solution specified in our problem. The sizes of the categories vary. or not. The parking lots of the larger categories may also be used by smaller ones. The parking lots of smaller categories are off- limits to larger categories. MDVSP-based models usually ∑︁ ∑︁ ∑︁ ∑︁ handle different vehicle categories by incorporating vehicle min 𝑑 𝑑 𝑑 𝑐 𝑦 𝑠 𝑠 + 𝑑 𝑐 𝑥 𝑒 𝑒 𝑑 𝑑 types into physical depots. This means that the number 𝑑 𝐷 𝑠 𝑆 𝑑 𝐷 𝑒 𝐴 ∈ ∈ ∈ ∈ of depots in the theoretical model will be a multiple of Subject To the number of physical depots, as their number should be multiplied by the number of vehicle categories. Depots ∑︁ = 1 (1) 𝑑 ∀ ∈ 𝑥 , 𝑢 𝑈 𝑒 defined in this way can also be called logical depots. 𝑑 + ∈ 𝐷 𝑢 ,𝑒 ∈ 𝑢 Based on [3] we present shortly how the model handles 𝑑 ∑︁ ∑︁ the depots. We use the following notations in the descrip- 𝑥 𝑥 𝑣 𝑉 𝑑 𝐷 , 𝑒 − 𝑑 𝑑 = 0 ∀ ∈ ∀ ∈ , , 𝑒 𝑑 tion: 𝑒∈𝑣 + 𝑒 ∈𝑣 − 𝑑 𝑑 (2) 𝐺 𝑉 , 𝐴 = () The graph used for the representation of 𝑡 + 𝑒 𝑡𝑣 𝛿𝑒 𝑥𝑒 𝐿, ≥ + − ( 1 − ) the problem. (3) 𝐷 The set of depots. 𝑈 The set of trips, represented by the nodes ∈ \ { ( ) | ∈ } ∀ ∈ (4) ∀ + 𝑣 𝑉 𝑎𝑡 𝑑 𝑑 𝐷 , 𝑒 𝑣 in 𝐺. 𝑡𝑒− + 𝛿𝑒 + 𝑥𝑒 𝐿 ≤ 𝑟𝑑 + 𝐿 (5) 𝐴 𝑑 𝐷 − The set of arcs belonging to depot ∈ . ∀ ∈ { ( ) | ∈ } ∀ ∈ (6) 𝑑 𝑣 𝑎𝑡 𝑑 𝑑 𝐷 , 𝑒 𝑣 𝑑 The set of arcs which should be covered 𝐴 𝑞 ∑︁ by both vehicle and driver ( 𝑑 𝑥 𝑘 𝑒 ≤ 𝑑 𝐴 𝐴 ) 𝑞 𝑑 ⊆ , 𝑑 𝐷 , . 𝑑 ∀ ∈ (7) 𝑉𝑑 The set of trip vertices belonging to depot 𝑒 𝑎𝑡 𝑑 ∈ ( ) − ∑︁ ∑︁ 𝑑 𝐷 𝑦 𝑣 𝑉 , 𝑥 𝑑 , 𝐷 . − = 0 ∀ ∈ ∀ ∈ ∈ 𝑑 𝑑 𝑠 𝑒 𝑑 𝐷𝑢 𝑢 𝑠 ∈ + The set of depots, from which can be 𝑆 𝑣 𝑑 ( 𝑣 ) 𝑒 ∈ 𝑑 served. (8) + The set of all outgoing arcs of vertex ∈ 𝑣 𝑣 𝑉 ∑︁ 𝑑 𝑑 𝑑 − = 0 ∀ ∈ ∀ ∈ 𝑦 𝑥 , 𝑒 𝐴 𝑑 𝐷 . , in 𝑠 𝑒 𝑞 𝐺 . − 𝑠 ∈𝑆𝑑 (𝑒) The set of all incoming arcs of vertex 𝑣 𝑣 ∈ (9) 𝑉 𝐺 in. ∑︁ 𝑒 + 𝑑 𝑥 The head vertex of 𝑒 ∈ 𝐴 in 𝐺 . 𝑒 ≤ 𝑑 𝑝 , ℎ ( 𝑇 ) (10) 𝑒 − 𝑑 𝑒 The tail vertex of ∈ 𝐴 in 𝐺 . 𝑒 ∈ 𝐴 (𝑇 ) ℎ 𝑎𝑡 𝑑 𝑑 𝐷 𝐺 ( ) The arrival vertex of depot ∈ in. ∀ 𝑇 ∈ {𝑇 , ..., 𝑇 1 (11) 𝑚 } ∀ ∈ ∀ ∈ , ℎ 𝐻 , 𝑑 𝐷 𝑡 𝑣 The running distance up to trip vertex 𝑣 in its vehicle schedule. 𝑥 , 𝑦 𝑒𝑠 ∈ { 𝑑 𝑑0 1} ≥ 0 ∀ ∈ ∀ ∈ , , 𝑡 , 𝑒 𝐴 , 𝑠 𝑆 , 𝑣 𝑑 𝑑 𝐿 (12) A constant larger than the longest possible running distance. ∀𝑑 ∈ 𝐷 , ∀𝑣 ∈ 𝑉 \ {𝑎𝑡 (𝑑) | 𝑑 ∈ 𝐷 (13) 𝛿 𝑒 The running distance of arc . 𝑒 𝑟𝑑 The maximal allowed running distance for 4 Conclusions the vehicles in depot 𝑑 ∈ 𝐷. In this paper, we reviewed how the optimization model 𝑘 𝑑 The number of vehicles available in depot solving the combined vehicle and driver scheduling problem 𝑑 𝐷 ∈ . was implemented in practice. We presented what specific 𝑆 The set of valid driver schedules generated 𝑑 developments were necessary in order to be able to take for depot 𝑑 ∈ 𝐷 into account all the practical requirements that arose. We 𝑆 𝑣 ( ) The set of driver schedules containing the 𝑑 presented what implementation technologies we would trip vertex 𝑣 ∈ 𝑉𝑑 (𝑆𝑑 (𝑢) ⊆ 𝑆𝑑). use to facilitate the solvability of the problem for critical 𝑇 , ..., 𝑇 1 The time slots, for which parking capaci- 𝑚 inputs, when in the basic case there was no solution, or the ties should be checked. methods used for the solution proved to be too slow. We 𝐻 The set of those stations and parking lo- performed tests on real problems to verify the effectiveness cations that can be used by the vehicles. 𝑑 of the technologies. 𝐴 𝑇 ( ) The set of those arcs that covers time slot ℎ 𝑇 ℎ 𝐻 𝐴 𝑇 𝐴 at location ∈ ( ( ) ⊆ ). References 𝑑 ℎ 𝑑 𝑑 ( ) The number of parking places at location 𝑝 𝑇 ℎ [1] Adler, J.D., Mirchandani P.B. (2016). The vehicle sched- in time slot for vehicles ℎ 𝑇 𝑇 , ..., 𝑇 ∈ { 1 } 𝑚 of depot 𝑑 ∈ 𝐷. cles.Transportation Science, 51(2), 441–456, 2016. uling problem for fleets with alternative-fuel vehi- 𝑆 𝑒 () The set of driver schedules containing the 𝑑 [2] Békési, J., Brodnik, A., Krész, M., and Pas, D. (2009). An Inte- arc 𝑒 ∈ 𝐴𝑑 (𝑆𝑑 (𝑒 ) ⊆ 𝑆𝑑 ). grated Framework for Bus Logistics Management: Case Stud- 𝑑 17 𝑐 𝑠 𝑆 The cost of schedule ∈ . ies, In: S. Voss, J. Pahl and S. Schwarze (eds.), Logistik Man-𝑠 𝑑 𝑑 The cost of arc ∈ . agement: Systeme, Methoden, Integration, Springer, 389–411. 𝑐 𝑒 𝐴 𝑒 𝑑 MATCOS-25, October 9–10,2025, Koper, Slovenia Viktor Árgilán, József Békési, Gábor Galambos, and Imre Papp [3] Békési, J., Nagy, A. (2020). Combined Vehicle and Driver Sched- [10] Kliewer, N., Mellouli, T. and Suhl, L. (2006). A time-space uling with Fuel Consumption and Parking Constraints: a Case network based exact optimization model for multi-depot bus Study. Acta Polytechnica Hungarica, 17(7): 45–65. scheduling, European Journal of Operational Research, 175: [4] Bodin, L., Golden, B., Assad, A. and Ball, M. (1983). Routing 1616–1627. and Scheduling of Vehicles and Crews: The State of the Art. [11] Li, J-Q. (2013). Transit bus scheduling with limited energy, Computers and Operations Research, 10: 63–211. Transportation Science, 48(4), 521–539. [5] Freling, R., Huisman, D. and Wagelmans, A.P.M. (2003). Models [12] Löbel, A. (1997). Optimal Vehicle Scheduling in Public Transit, and algorithms for integration of vehicle and crew scheduling. Ph.D. thesis, Technische Universitaet at Berlin. Journal of Scheduling, 6: 63–85. [13] Mesquita, M., Moz, M., Paias, A., Paixao, J., Pato, M. and Re- [6] Haase, K., Desaulniers, G. and Desrosiers, J. (2001). Simultane- spício, A. (2011), A new model for the integrated vehicle-crew- ous vehicle and crew scheduling in urban mass transit systems, rostering problem and a computational study on rosters, Jour- Transportation Science, 35(3): 286–303. nal of Scheduling,14, pp 319–334. [7] Hadjar, A., Marcotte, O. and Soumis, F. (2001). A Branch-and- [14] Ribeiro, C.C., Soumis, F. (1994). A Column Generation Ap- Cut Algorithm for the Multiple Depot Vehicle Scheduling proach to the Multiple-Depot Vehicle Scheduling Problem. Problem. Tech. Rept. G–2001–25, Les Cahiers du Gerad, Mon- Operations Research, 42(1): 41–52. treal. [15] Steinzen, I. (2007), Topics in integrated vehicle and crew sched- [8] Horváth, M., Kis, T. (2019), Computing strong lower and upper uling in public transit. PhD thesis, University of Paderborn. bounds for the integrated multiple-depot vehicle and crew [16] Steinzen, I., Gintner, V., Suhl, L. and Kliewer, N. (2010), A scheduling problem with branch-and-price. Central European Time-Space Network Approach for the Integrated Vehicle- Journal of Operations Research, 27(1), pp 39–67. and Crew-Scheduling Problem with Multiple Depots, Trans- [9] Huisman, D., Freling, R. and Wagelmans, A.P.M. (2005). portation Science, 44, pp 367–382. Multiple-depot integrated vehicle and crew scheduling. Trans- portation Science, 39: 491–502. 18 ALGatorGraph: A Java Library for Graph Generation and Manipulation within the ALGator System Boštjan Hren and Tomaž Dobravec University of Ljubljana Faculty of Computer and Information Science Ljubljana, Slovenia Figure 1: The DIRECTED_SCALE_FREE and LAYERED graphs generated with the ALGatorGraph library. Abstract of purely theoretical analysis. In response, several complemen- In this paper, we present the ALGatorGraph Java library, an ex- tary methodologies have emerged—experimental algorithmics, al- tension of the ALGator framework for graph-related problems. It gorithm engineering, and the broader empirical approach—each provides a unified interface for graph data structures and supports contributing unique perspectives and tools for studying algorithms generation of diverse graphs. A Maximum Flow problem case study in practice. shows how easily algorithms can be implemented and experimen- Experimental algorithmics [3] focuses on the empirical study of al- tally evaluated within ALGator using the ALGatorGraph library. gorithms through computational experiments. Borrowing from the scientific method, it emphasizes reproducibility, hypothesis testing, CCS Concepts and data-driven insights. Rather than proving abstract bounds, re- searchers design experiments to observe algorithm behavior across • Theory of computation → Algorithm design techniques; Graph algorithms analysis; Approximation algorithms analysis; Data helps to uncover performance trends, identify bottlenecks, and a wide range of inputs, environments, and configurations. This structures design and analysis. challenge theoretical assumptions with real-world evidence. Keywords grates design, analysis, implementation, and experimental evalu- Closely related, but distinct, is algorithm engineering, which inte- Experimentation, algorithm evaluation, graph problems ation into a cohesive development cycle [5]. Treating algorithms as both mathematical objects and software components, algorithm 1 engineering emphasizes modular design, rigorous implementation, Introduction and iterative refinement, often leveraging hardware characteris- The development and analysis of algorithms has long been a central tics and system-level concerns to improve performance. In parallel, theme in computer science, traditionally rooted in rigorous theo- the empirical approach provides a broader umbrella, extending retical frameworks. Theoretical computer science provides formal to benchmarking, comparative evaluation, and large-scale testing, guarantees about algorithmic correctness, complexity, and optimal- prioritizing observation and measurement over formal proofs. ity, often under idealized assumptions [1]. While this foundation While these methodologies differ in emphasis, they share a commit- remains indispensable, the increasing complexity of computational ment to complementing theoretical analysis with practical evidence. environments and problem domains has revealed the limitations 19 B. Hren and T. Dobravec Experimental results not only validate formal claims but also ex- The GraphCreator class also includes the method called import- pose behaviors, trade-offs, and phenomena that purely theoretical GraphsFromFolder(), which enables reading graphs from various reasoning may overlook. In this landscape, dedicated tools and file formats, including popular formats such as GraphML, CSV, DOT, frameworks play a critical role: they provide researchers with envi- and JSON. This functionality is essential for integrating external ronments in which reproducibility, transparency, and fairness of data sources into the ALGator system, allowing algorithm testing experiments are guaranteed. on real-world networks and graphs. The importer automatically ALGator is one such framework [2]. Designed for the systematic detects the file format and appropriately processes both the graph evaluation of algorithms, it enables researchers to compose testing structure and any edge weights, providing a high degree of flexibil- environments in which multiple algorithms can be implemented, ity when working with diverse input data. The function returns the tested, and compared across diverse test cases and test sets. By data as JGraphT objects. JGraphT is a robust library that serves as ensuring repetition, fair judgment, transparency, and the public the foundation for the classes we have developed and implemented. availability of results, ALGator supports rigorous experimental We selected it as the most effective library in the graph domain, practice while fostering collaboration and reproducibility in the and it is therefore used to facilitate graph import. research community. The ALGatorGraph package also includes utility functions for graph Building on this foundation, ALGatorGraph extends ALGator with description, where the getGraphDescription method provides a native support for directed and undirected graphs, a central do- human-readable summary of the properties and parameters for each main in algorithmics. Many of the most challenging and widely graph type, significantly facilitating system usage. The package studied algorithmic problems—from shortest paths and network is designed to simplify the integration of graph algorithms into flows to clustering and graph isomorphism—are inherently graph- the ALGator system while maintaining a high degree of flexibility based. By facilitating the use of graphs as both input and output and adaptability for various types of analyses. The combination data structures, ALGatorGraph lowers the barrier for integrating of automated graph generation, import capabilities, and robust graph-related projects into the experimental environment of ALGa- methods for converting graphs from files or JGraphT objects makes tor. This allows researchers not only to implement and test graph this package an invaluable tool for developers aiming to test and algorithms more easily but also to benefit from the reproducible compare algorithms on graphs within a controlled environment. and transparent evaluation framework that ALGator provides. GraphConverter. The GraphConverter class enables the con- With ALGator and ALGatorGraph, experimental algorithmics is version of graphs from various JGraphT formats into ALGator’s equipped with a robust infrastructure for studying graph algorithms native graph representations. This class provides methods for con- under controlled and repeatable conditions. This integration high- verting both directed and undirected graphs, supporting both simple lights the importance of aligning theoretical insights, empirical edges (DefaultEdge) and weighted edges (DefaultWeightedEdge). experimentation, and practical implementation—ultimately advanc- Special attention is given to the robust handling of different node ing the broader field of experimental algorithmics. types: the class automatically recognizes and converts both numeric and string nodes into a unified integer format, ensuring consistency 2 when working with algorithms. With this converter, nearly any The ALGatorGraph library The library is a new ALGator library that repre- ALGatorGraph graph representation can be imported into ALGator for algorithm testing. sents a key component for working with graphs within the system. The library extends ALGator’s core functionalities by providing specialized tools for graph construction, transformation between different representations, and graph management within the ALGa- 3 Graph data structure tor framework. The ALGatorGraph library builds upon the JGraphT The classes in the ALGatorGraph package support working with library [4], leveraging its core data structures while extending es- both directed and undirected graphs. The DEdge class rep- sential functionality for constructing and managing graphs within resents a directed edge with a source vertex, a target vertex, and a the ALGator testing environment. The primary components en- weight, where the order of vertices is significant. It is used together abling graph construction and manipulation are the GraphCreator with the DGraph class, which implements a directed graph and GraphConverter classes. using adjacency lists (via a HashMap) and supports generic types GraphCreator . This class provides a comprehensive set of capa- for vertices (V) and weights (W). Methods include adding/remov- bilities for generating graphs with diverse topologies. It supports ing vertices and edges, as well as querying neighbors, with time over 50 different types of graphs, ranging from basic structures complexities indicated in the comments (e.g., O(1) for adding a such as complete and bipartite graphs to more complex forms, in- vertex). cluding Petersen graphs, lattices, hypercubes, and various special For undirected graphs, the Edge and Graph classes graphs (e.g., the diamond graph, the Buckminsterfullerene graph, are provided. The Edge class stores two vertices (vertex1, vertex2) or the Zachary’s Karate Club graph). Each graph type is accessible and a weight, where the vertex order is irrelevant (e.g., the edge through a simple interface, allowing the user to specify the desired A–B is equivalent to B–A). The Graph class implements an undi- structure using an enum value along with relevant parameters. The rected graph using adjacency lists, where adding an edge A–B auto- class also supports random graph generation according to various matically adds B–A. Both classes override the equals(), hashCode(), models, including the Barabási–Albert model, the Watts–Strogatz and toString() methods and provide operations similar to their small-world model, and the Erdős–Rényi model of random graphs. directed counterparts, but adapted for undirected edges. 20 ALGatorGraph: A Java Library for Graph Generation and Manipulation within the ALGator System The main difference lies in directionality: DEdge/DGraph require an 0.3 , 0.9 , 0.1 , 0.2 , edges , no des }; explicit direction (e.g., for paths), whereas Edge Gr aph gr aph = G r a p h C r e a t o r . g e n e r a t e G r a p h ( args ); / Graph treat edges as symmetric (e.g., for friendship networks). Both graph types are • Type4 generator generates a layered graph with the given optimized for fast adjacency-list operations and are compatible layerCount and nodesPerLayer parameters. In this case, with the JGraphT library. a method createLayeredGraph(int layerCount, int nodesPerLayer, double connectionProbability) is 4 Usage Example: The MaxFlow Project used. In all the cases the probability of creating edges be- To illustrate the usage of the ALGatorGraph library, we developed a tween vertices in consecutive layers, connectionProbability, simple example project focused on the Maximum Flow problem [6]. is set to 0.6. The primary goal of this project is not to provide a comprehensive • Type2 and Type5 generators receives the folderName and analysis or optimization of the problem itself, but rather to demon- the graphNumber parameters and import the corresponding strate how a project can be structured around graph-based data. In graph from the given folder using the GraphCreator.import- this example, the core data structure is a graph, implemented as an GraphsFromFolder() method.. object from the ALGatorGraph library, highlighting how the library Input. A Java class used to store the input data of a test case facilitates graph creation, manipulation, and integration within a contains a triple (graph, source, sink) and is provided to algorithms computational workflow. This approach showcases the practical as an argument. application of ALGatorGraph for managing complex network struc- tures and serves as a template for building more sophisticated p ub l i c class In put e x t e n d s A b s t r a c t I n p u t { DGraph < Integer , Integer > gra ph ; graph-based projects in the ALGator environment. int s o u r c e ; To use the ALGatorGraph library in the project, the correspond- int sink ; ing JAR packages must be imported, thereby providing access to } the classes and methods within the ALGatorGraph package. This Output. Output contains an integer representing the maximum is accomplished by configuring the ALGator ProjectJAR array flow; this object is returned by the algorithm. variable. In the following we list the properties that were defined in the project. p ub l i c class O u t p u t e x t e n d s A b s t r a c t O u t p u t { int m a x F l o w ; Parameters. The names and types of the parameters required to } generate input for the test. Source (int) and Sink (int) are used in every test to specify the source and sink vertices in the Indicators. In our project, we have an indicator called Check, graph. and Edges (int) are used for generating Nodes (int) which verifies whether the algorithm’s output is correct by per- DIRECTED_SCALE_FREE forming a simple comparison with the expected output. The method graphs, determining the number of vertices and edges in the graph. and NodesPerLayer (int) Layers (int) IndicatorTest¸_Check returns a result of "OK" or "NOK" in case are used when generating a graph to specify the number LAYERED of failure. Additional indicators of algorithm success can also be of layers and the number of vertices per layer. For examples of both added, returning more complex objects if needed. types of graphs, see Figure 1. Folder (String) provides the path p ub l i c class I n d i c a t o r T e s t _ C h e c k e x t e n d s to the directory containing graph files; in our case, these are the A b s t r a c t I n d i c a t o r T e s t < TestCase , Output > { graphs and layered_graphs folders within the project directory. @ O v e r r i d e NumOfGraph (int) p ub l i c O b j e c t g e t V a l u e ( T e s t C a s e tc , O u t p u t o u t p u t ) { is used when the folder contains multiple graph files, indicating which graph to use in sequence. r et u r n o u t p u t . m a x F l o w Generators. == In the project we defined several generators to provide tc . g e t E x p e c t e d O u t p u t (). m a x F l o w ? " OK " : " NOK " ; different types of tests. } • } Type0 and Type3 generators produce explicitly specified graphs that are used for trivial testing. Graphs are created Algorithms. In the project, we implemented two algorithms: using a simple API; for example: • Edmonds–Karp implementation of the Ford–Fulkerson al- DGraph < Integer , Integer > gra ph = new DGraph < >(); gorithm, which uses BFS to find augmenting paths and for ( int i = 0; i < 6; i ++) { achieves a time complexity of O(nm2). gr aph . a d d V e r t e x ( i ); } • Dinic’s algorithm, which combines BFS and DFS to compute gr aph . a d d E d g e (0 , 1). s e t W e i g h t (3); the maximum flow with a time complexity of O(n2m). // ... According to ALGator’s convention, an algorithm is represented The only inputs to the generator are the Source and Sink by a class that implements an execute() method, which receives parameters, allowing a limited set of basic tests to be per- an Input object as a parameter and returns an Output object as the formed. result, as shown in the following listings. • Type1 generator receives additional nodes and edges pa- // An i m p l e m e n t a t i o n of the E d m o n d s K a r p a l g o r i t h m rameters and generates a random graph with a specified p ub l i c class A l g o r i t h m e x t e n d s P r o j e c t A b s t r a c t A l g o r i t h m { number of vertices and edges. p r o t e c t e d O u t p u t e x e c u t e ( Inp ut in put ) { S t r i n g [] args = { " D I R E C T E D _ S C A L E _ F R E E ", 21 B. Hren and T. Dobravec Figure 2: Comparison of the complexities of the Edmonds-Karp and Dinic algorithms on (a) DIRECTED_SCALE_FREE graphs (upper chart) and (b) LAYERED graphs (lower chart). DGraph < Integer , Integer > gra ph = inp ut . gra ph ; than Edmonds–Karp algorithm (see Figure 2, bottom chart). The int s o u r c e = inp ut . s o u r c e ; average speedup in this scenario is 2.9x. int sink = i npu t . sink ; int m a x F l o w = e d m o n d s K a r p ( graph , source , sink ); 5 Conclusions O u t p u t o u t p u t = new O u t p u t ( m a x F l o w ); The ALGatorGraph Java library has been developed as a supporting r e tu r n o u t p u t ; tool for ALGator projects that address graph-related problems. Its } main benefits are twofold: it offers a unified interface for imple- } menting graph data structures, and it provides the capability to Testsets. generate a wide variety of graphs—both through parameter-based In the project, several test sets were implemented. The first test set contains directed scale-free graphs generated by a generation and by reading graphs from the library. Type1 Through the MaxFlow project we have demonstrated how straight- test set generator. The test cases were created using the following for-loop–based description: forward it is to implement graph-related problems in ALGator using ALGatorGraph. The results obtained in the experiments highlight $for {i ,1 ,30 ,1}: T ype 1 ::0: $ {10 0* i -1}: $ {10 0* i }: $ {50 0* i } some characteristics of the two implemented algorithms, but their specific values are not the main point. The real significance lies This generates 30 Type1 test cases with parameters 𝑛 = 100 ∗ 𝑖, in how easily such results can be obtained by combining ALGa- 𝑚 𝑖 𝑖 𝑖 , , . . . , = 500 ∗ , source = 0, and sink = 100 − 1, for = 1 2 30. torGraph’s ability to handle graph data structures with ALGator’s Running the algorithms on this test set yields the results shown functionality for conducting and analyzing experiments. in Figure 2 (upper chart). For this configuration, the complexity of both algorithms increases with the number of nodes. On aver- References age, the Edmonds–Karp algorithm performs about 33% faster than [1] S. Arora and B. Barak. 2009. Computational Complexity: A Modern Approach. Dinic’s algorithm on this test set. Further tests revealed that the Cambridge University Press. advantage of the Edmonds–Karp algorithm on directed scale-free [2] Tomaž Dobravec. 2025. ALGator: Online Implementation of the System for graphs is greater when edges are evenly distibuted. As the disparity Automated Algorithm Execution and Evaluation. (2025). https://algator.fri.uni- lj.si between node degrees increases, this advantage diminishes, and [3] Catherine C. McGeoch. 2012. A Guide to Experimental Algorithmics. Cambridge in some cases Dinic’s algorithm even outperforms Edmonds–Karp University Press. algorithm. [4] Dimitrios Michail, Joris Kinable, Barak Naveh, and John V. Sichi. 2020. JGraphT–A Java Library for Graph Data Structures and Algorithms. ACM Trans. Math. Softw. Another test set used in this project consists of layered graphs 46, 2, Article 16 (May 2020), 29 pages. generated by a [5] Matthias Müller-Hannemann. 2010. Algorithm Engineering, Bridging the Gap Type4 test set generator. The test cases were defined Between Algorithm Theory and Practice. Springer-Verlag Berlin and Heidelberg using the following for-loop–based description: GmbH & Co. K. [6] Ruiheng Zhang. 2024. Network Flow of Graph Theory and Its Application. In $for {i ,1 ,30 ,1}: T ype 4 ::0: $ {( i *3 -2)*( i +5)+2 -1}: $ { i *3}: $ { i +5} Proceedings of the 1st International Conference on Engineering Management, In- formation Technology and Intelligence - EMITI. INSTICC, SciTePress, 692–696. This generates layered graphs with 3 ∗ 𝑖 layers and 𝑖 https://doi.org/10.5220/0012968800004508 + 5 nodes per layer, for 𝑖 = 1, 2, . . . , 30. Running both algorithms on this test set shows that, for these layered graphs, Dinic’s algorithm is faster 22 Engineering CSFLOC: A Subsumption-Driven Clause-Counting SAT Solver Gábor Kusper kusper.gabor@uni- eszterhazy.hu Eszterházy Károly Catholic University Eger, Hungary Abstract This paper briefly recalls CSFLOC and focuses on an optimized Java implementation. Clause counting, a technique from the #SAT domain, offers a complementary angle to conflict-driven clause learning (CDCL). We revisit CSFLOC ( 1.1 Related work Counting Subsumed Full-Length Ordered Clauses) and present an engineered implementation that turns Modern SAT solving is dominated by conflict-driven clause learn- its theoretical "last-1-bit" observation into practical speedups. ing (CDCL) with powerful preprocessing and restart heuristics; The solver represents full-length clauses with a binary counter see the updated for a broad survey of Handbook of Satisfiability and performs sign-aware bucket scans keyed by the last-literal techniques and applications [4]. Alongside CDCL, a classical but index, allowing safe jumps that skip a block of consecutive full- distinct line is and inclusion–exclusion–based clause counting length clauses subsumed by input clauses. We map the known reasoning, which reasons about sets of full-length (maximal) pseudocode of CSFLOC line-by-line to Java, detailing the data clauses and their subsumption structure. structures that make the inner loop fast: POS/NEG buckets per Early work by Iwama established counting-style satisfiability index, small effected/learned caches, and carry-like counter up- tests with average-case guarantees [11]. Lozinskii developed ex- 𝑛 𝑗 − dates that implement jumps of size 2 without big integers. act propositional model counting (the so-called #SAT problem) Preordering (variable renaming) strategies—composable flags in [15], while Birnbaum–Lozinskii connected counting tightly to B,C,H,I,R,S,W—shape the distribution of last-literal indices and, Davis–Putnam–style branching [2]. A complementary strand via an island/strait view of the counter, bias the solver toward is Andrei’s “inverting resolution,” which frames satisfiability longer trailing 1-blocks and shorter 0-gaps. A concise empirical counting via inverse propositional resolution and normaliza- snapshot shows where CSFLOC is competitive (over-constrained, tion [1]. These works already emphasize that ordering, last-literal dense-structure instances) and where CDCL remains preferable. positions, and subsumption can unlock substantial contiguous Source code is available at . “jumps” in the enumeration space. http://fmv.ektf.hu/tools.html The inclusion–exclusion principle has been repeatedly ex- Keywords plored for model counting and even SAT itself. Bennett and Sankaranarayanan proposed an inclusion–exclusion counter with CSFLOC, SAT solving, clause counting subsumption pruning for 𝑘-SAT [3], while Zaleski implemented 1 a SAT solver in Maple using inclusion–exclusion and Bonferroni Introduction inequalities [16]. More recently, inclusion–exclusion has been The propositional satisfiability problem (SAT) asks whether a combined with dynamic programming over small treewidth for propositional CNF formula has an assignment of truth values to projected model counting (PMC), yielding practically competitive its variables that makes the formula true. SAT is one of the most- PMC/#SAT solvers [8]. These results corroborate the general mes- researched NP-complete [7] problems in computer science, with sage behind CSFLOC: structural regularities (e.g., small treewidth, applications ranging from theoretical computer science and arti- strong subsumption) can be exploited to skip large contiguous ficial intelligence to hardware design and formal verification [4]. blocks in the search space. Modern SAT solvers build on conflict-driven clause learning CSFLOC itself belongs to the full-length clause–counting fam- (CDCL), an extension of the classical Davis Putnam Logemann Optimized CCC ily. It is the successor of the algorithm [14]. It Loveland (DPLL) procedure [9] that adds conflict analysis, clause uses a counter to count subsumed full-length clauses. By studying learning, and non-chronological backtracking (backjumping) [4]. Optimized CCC we observed that its full-length clause counter An interesting question is how many models a SAT instance can be increased on its last 1 bit in the best case [13]. CSFLOC is has; this is the #SAT problem [11, 15, 3], i.e., counting the satis- based on this observation. It also uses a data structure in which fying assignments of a CNF formula [10]. the clauses are ordered by the index of their last literal. These CSFLOC ( ) is Counting Subsumed Full-Length Ordered Clauses two improvements result in a faster algorithm which can com- a clause-counting approach that enumerates full-length clauses pete with a state-of-the-art SAT solver on problems with lots of via a binary counter and uses subsumption to jump over blocks. clauses, like Black-and-White 2-SAT problems [6] and weakly It was introduced in [14, 13]. Although CSFLOC is a general nondecisive SAT problems [5]. SAT solver, it is inspired by classical #SAT techniques such as full-length clause counting. 1.2 Revisiting the CSFLOC algorithm CSFLOC was introduced in [13]. In this subsection, we recall its Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or pseudocode and the main theoretical results. distributed for profit or commercial advantage and that copies bear this notice and The following properties underpin Algorithm 1; we state them the full citation on the first page. Copyrights for third-party components of this without proof, since these theorems are already proven in [13]. work must be honored. For all other uses, contact the owner /author(s). MATCOS-25, 9–10 October 2024, Koper,Slovenia Lemma 1 (Observation 2.). In the inner loop of CSFLOC, if © 2025 Copyright held by the owner/author(s). 𝑘 𝐶 𝑗 = IndexOfLastPositiveLiteral( ) and is the last-literal index of a 23 MATCOS-25, 9–10 October 2024, Koper,Slovenia Gábor Kusper Algorithm 1 CSFLOC(𝑆) Buckets 𝑆 [1..𝑛] (lines 1–2): built by HighLevelReader us- Require: ing two Clause arrays 𝑆 is a non-empty list of ordered clauses with variable index function • clauseListOrderedByLastVarIndexPos[i] and 𝐼 . Ensure: • clauseListOrderedByLastVarIndexNeg[i] and 𝑖 If 𝑆 is satisfiable it returns a solution of 𝑆 , otherwise ∈ returns the empty set. {1..𝑛}, 1: 𝑛 : number of variables in 𝑆; = storing clauses with IndexOfLastLiteral 𝐶 𝑖 whose last ( ) = 2: 𝑆 𝑖 : 𝐶 𝐶 𝑆 IndexOfLastLiteral( 𝐶 ) 𝑖 , where 𝑖 1..𝑛; ] = { | ∈ ∧ = } = [ literal is positive / negative. 3: Counter init (line 3): count is represented by the Boolean 𝑐𝑜𝑢𝑛𝑡 : = 0; 4: 𝑐𝑜𝑢𝑛𝑡 < 2 while counter[1..n] do 𝑛 vector (1 = positive, 0 = negative). Im- 5: 𝑖𝑛𝑐𝑟 𝑒𝑚𝑒𝑛𝑡 : 0; best black = plementation uses a practical initial value: the 6: clause; it sets the first 1-bit and calls 𝐶 : = FullLengthClauseRepresentationOf( 𝑐𝑜𝑢𝑛𝑡 ); addEffectedClause 7: 𝑗 : IndexOfLastPositiveLiteral(𝐶); 𝑗 < 𝑛; 𝑗 : 𝑗 1 = = = + for to prime caches. do Full-length clause 𝐶 (lines 4–6): Not materialized: the cur- 8: 𝐷 𝑆 𝑗 such that 𝐷 subsumes 𝐶 ∃ ∈ [ ] then if rent encodes 𝐶; subsumption is tested directly counter 9: 𝑖𝑛𝑐𝑟 𝑒𝑚𝑒𝑛𝑡 : 2 ; = 𝑛 𝑗 via methods like . − Clause.subsumedBy(counter) 10: Setting 𝑗 (line 7): Computed as the index of the last 1-bit 𝑗 : = 𝑛 + 1; 11: end if in ; passed as to counter index usingBestClause_v4( 12: index, counter) end for . 13: Scan 𝑗 ..𝑛 for a subsumer (line 7): One of the main meth- if 𝑖𝑛𝑐𝑟 𝑒𝑚𝑒𝑛𝑡 = 0 then 14: return 𝐶; ¬ ods performs a tra-usingBestClause_v4(i) sign-aware 15: else versal. It tries to find a subsumer clause by testing 16: 𝑐𝑜𝑢𝑛𝑡 : 𝑐𝑜𝑢𝑛𝑡 𝑖𝑛𝑐𝑟 𝑒𝑚𝑒𝑛𝑡 ; = • + effected/oldEffected at level 𝑖, 17: • learnedClausesPos at level 𝑖 end if 18: • clauseListOrderedByLastVarIndexPos at level 𝑖, end while 19: return ; {} • clauseListOrderedByLastVarIndexNeg from level 𝑖 + 1 in a loop. This realizes the theoretical 𝑗 ..𝑛 loop once the sign pattern subsuming clause 𝐷 ∈ 𝑆 [ 𝑗 ], then 𝑗 ≥ 𝑘, hence the largest possible of 𝐶 is taken into account. jump is 𝑛 −𝑘 2 . Subsumption test (line 8): Bit-level subsumption check is done by . D.subsumedBy(counter) Theorem 1 (Soundness and Completeness of CSFLOC). 𝑛 𝑗 Increment by (line 9): − 2 is done by the other main func- Algorithm 1 is sound and complete. tion increaseCounter_v5 which executes the jump with carry-like bit flips on , equivalent to adding 2 counter 𝑛 𝑗 − The 2. Observation Lemma, i.e., Lemma 1 states that in case 𝑛 𝑘 to the integer counter; hooks update caches and may insert − of CSFLOC the largest possible jump is 2 , and Theorem 1 learned clauses via . addEffectedClause states that this trick is valid, i.e., CSFLOC is a general SAT solver algorithm. See the proof of these theoretical results in [13]. In Return ¬𝐶 (line 14): Complement the counter and emit the model (optionally map the model back to the original this paper we discuss the practical implementation of CSFLOC. variable indices if a renaming was applied). 2 Pseudocode-to-Java Mapping Termination (lines 15–19): The outer loop in the method CSFLOC_v7() repeats until a model is returned or the The Java implementation referenced in this paper is available jumps exhaust 0, 2 , in which case UNSAT is reported. [ ) 𝑛 at . This section http://fmv.ektf.hu/files/CSFLOC19.java is highly technical. We suggest consulting the Java source code to understand this section. 2.1 Entry Points and Call Flow 2.3 Supporting Data Structures and Caches The Java implementation uses the following functions and flow • Sign-aware buckets [ ] : In the pseudocode 𝑆 𝑗 denotes the of calls: set of clauses whose last-literal index is 𝑗 . In the imple- • main calls + mentation we maintain two arrays per index, 𝑆 [ 𝑗 ] and • DIMACSReader (parse DIMACS CNF), which calls − 𝑆 [ 𝑗 ], for clauses whose last literal is positive or negative, • HighLevelReader (variable ordering, bucket build), which respectively; these are the Java arrays calls – clauseListOrderedByLastVarIndexPos[j] and • CSFLOCSolver.CSFLOC_v7() (outer loop), which calls – clauseListOrderedByLastVarIndexNeg[j]. • usingBestClause_v4 (inner scan), which calls + − The inner scan first probes 𝑆 [ 𝑗 ] and then 𝑆 [𝑘] for 𝑘 > 𝑗 , • increaseCounter_v5 ( jump) matching the positive/negative hit order. • the outer loop returns eventually a model or UNSAT; • Effected / oldEffected: per-index, per-sign small working • if a model is found, then main calls SimpleCheckSolution sets prioritized before base buckets. which validates the model. • Learned clauses : bounded per-index pools – learnedClausesPos, 2.2 Element-by-Element Mapping – learnedClausesNeg We map each step of Algorithm 1 with concrete classes and filled after successful subsumption hits; probed before base methods. buckets. 24 Engineering CSFLOC MATCOS-25, 9–10 October 2024, Koper,Slovenia • Best black clause fields: bestBlackClause is used to (e.g., {−1, 8, 23} becomes {−1, 25, 23}, moving the last literal from set the first index before the main loop to gain a small 23 to 25). Maintaining bucket memberships and all derived caches speed-up. at runtime would impose a prohibitive , so our solver burden • Renaming metadata: permutation 𝑡 𝑟 𝑎𝑛𝑠𝑙 𝑎𝑡 𝑒 maintained performs renaming only once before the main loop (see section 4). by (see section 4); used for bucket in- In conclusion, CSFLOC runs faster when the last island is HighLevelReader dices and for mapping models back. long and the straits between islands are short. Proving these observations in full generality is challenging; for random 3-SAT, 3 Counter Dynamics, Hits, Islands, and Straits however, parts of them can be established rigorously. A complete In this section, we introduce the intuitive notions of positive theoretical treatment remains an open line of research. hit The next section provides more information on variable re- negative hit islands straits , , and to help explain how CSFLOC naming strategies. works. Hit types. Let 𝐶 be the current full-length clause encoded by 4 Variable Renaming Strategies counter, and let 𝑘 = IndexOfLastPositiveLiteral(𝐶) denote the CSFLOC scans buckets from 𝑗 IndexOfLastPositiveLiteral 𝐶 = () index of the last 1-bit in 𝐶. For 𝑗 𝑘 , . . . , 𝑛 we say there is a ∈ { } upwards. Hence the determines the distribution variable order positive hit at level 𝑗 if there exists a clause 𝐷 ∈ 𝑆 [ 𝑗 ] whose last literal is positive and 𝐷 ⊆ of last-literal indices and which buckets are hit early. Renaming negative 𝐶 . For 𝑗 > 𝑘 we say there is a aims to (i) concentrate strong constraints early, (ii) lengthen hit at level 𝑗 if there exists a clause 𝐷 ∈ 𝑆 [ 𝑗 ] whose last literal is useful trailing 1-bit islands in the counter (larger jumps), and (iii) negative and 𝐷 ⊆ 𝐶 . Because positions greater than 𝑘 are 0s in improve cache locality. 𝐶, a positive last literal cannot subsume 𝐶 at those levels, hence only negative hits are possible above 𝑘. In both cases, CSFLOC Interface. variable-The solver accepts a composite string 𝑛 𝑗 − jumps by 2 (see Lemma 1). RenamingStrategy {B,C,H,I,R,S,W} consisting of letters from CSFLOC enumerates full-length clauses through a counter (uppercase or lowercase). Strategies are applied in the order of ap- which is a Boolean array (1=positive, 0=negative) and leverages pearance; lowercase variants indicate a milder weighting/priority. subsumption to jump. In the , the next iteration in the best case "IWCR" "HWCR" Typical presets in the implementation include: , , main loop increases the index of the by exactly 1, i.e., last 1-bit "BHWCR" "B" for random 3-SAT, and for pigeonhole families. the trailing 1-block grows by one. In the general case, adding 1 in base-2 may also flip higher 1-bits to 0 (carry), so a single step List of strategies and method names. can translate into a in the integer view of . large jump count B : renameBlackClauses. black Renames variables so that Islands and straits. clauses (all-negative) receive smaller last-literal indices. We interpret counter as a sea of zeros with Intuition: black clauses seed the formation of 1-bit is- occasional ones. A contiguous block of 1s is an ; a con-island tiguous block of 0s between islands is a lands in the counter; a long trailing island implies a high- strait . Let the rightmost island (closest to index 𝑛) be the . Growing the last last island probability large jump. Bringing black clauses forward reduces warm-up and favors long final islands or at least island by one (i.e., increasing the last 1-bit index by one) is the smaller straits. locally optimal step: it preserves previously gained structure and maximizes the chance of a larger safe jump because carries only W : renameWhiteClauses. Symmetric to B. It orders white affect positions to the left of the last island. clauses (all-positive). It reduces fragmentation caused by early all-positive buckets and helps separate positive hits When is a one-step growth possible? A one-step growth occurs from later negative scans. Used together with B to shape if there exists a clause 𝐷 whose last-literal index equals 𝑗 polarity structure. = IndexOfLastPositiveLiteral 𝐶 and 𝐷 𝐶 (a positive hit at level A clause is if it contains ( ) ⊆ S : renameStraitClauses. strait 𝑗 ). Otherwise the best we can hope for is that the next 1-bit sits exactly one negative literal (all others are positive). Such immediately to the left of 𝑗 (i.e., a negative hit at level 𝑗 1), so clauses tend to start a new island. This strategy assigns + the last island still grows by one. If the nearest 1-bit is farther left, a low index to the unique negative variable (and, if still the intervening 0s form a wider strait, which negatively affects unassigned, to the remaining variables of these clauses), the performance of CSFLOC. thereby shrinking the 0-gaps (straits) between islands and lowering last-literal positions. Implications for ordering and renaming. Because the inner scan I : renameIslandClauses. Symmetric to S (which targets starts at 𝑗 IndexOfLastPositiveLiteral 𝐶 and then checks for = ( ) clauses with exactly one literal), targets clauses negative I negative hits at𝑘 > 𝑗, we profit from (i) (long trailing large islands with exactly one literal (all others negative), i.e., positive 1-blocks) and (ii) between successive islands. This small straits definite Horn clauses. This strategy moves the unique aligns with runtime data: fewer, longer islands typically enable 𝑛 𝑗 last − positive variable to the largest index in the clause (its larger 2 jumps, whereas many short islands correlate with literal), but still assigns it a small global index, increasing smaller increments and more scans. Our preordering strategies the chance of positive hits and, in the island–strait view, aim to bias the distribution of last-literal indices accordingly; tending to cancel the last island. see the next section, especially the I (Island), S (Strait), and C H : renameDefiniteHornClauses. Uses the same syntac- (Clustering) variable-renaming strategies. tic filter as (clauses with exactly one positive literal), but I Why no dynamic renaming? One might attempt to dynamically is conservative: it processes a clause only if none of its lit- adjust the order so that the next 1-bit always follows the last one. erals has been renamed by any earlier strategy. Otherwise, However, the implementation reaches clauses by their it is the same as . Because triggers only on untouched last literal I H (a variant of watched-literal indexing). Swapping two variables clauses, it composes well with polarity-oriented strategies (say, 8 and 25) can change the last-literal index in many clauses such as (white) and (black); the combined effect is W B 25 MATCOS-25, 9–10 October 2024, Koper,Slovenia Gábor Kusper Table 1: Where CSFLOC excels vs struggles. Table 2: Qualitative impact of clustering on SATLIB. Instance type Best solver Intuition Suite no clustering cluster-2 cluster-3 Random SAT (uf50/uuf50) Glucose weak structure uf50 run time run time baseline ↓ ↓↓ Pigeonhole (small 𝑛) Glucose black and white uf75 often timeout many solved most solved ✓ ✓✓ WSN BW 2-SAT CSFLOC over-constrained uf100 timeout some solved many solved ✓ ✓✓ WnD UNSAT CSFLOC many clauses SM, dense >10% CSFLOC over-constrained index and sign. A simple clustering of variables into small con- SM, sparse <10% Glucose few subsumers tiguous groups (section 4) consistently improves run time on SATLIB: uf50 becomes notably faster; uf75/uf100, which tend to time out without clustering, become solvable under 2–3 variable to lower last-positive positions and stabilize long trailing islands. clusters. Frequency- and Horn-biased orders (R/H) further reduce R : simplerVariableRenaming. the last-positive indices encountered, increasing positive hits. It orders variables by their occurrence counts; high-frequency variables result in ear- Graph-induced encodings. On SM/BM/SBB encodings derived lier hits. from directed graphs, we observe a density threshold: for edge C : clusterVariables. Clustering orders variables so that densities above roughly 10%, CSFLOC tends to outperform Glu- variables that frequently co-occur in clauses are placed cose; below that, CDCL remains preferable. The effect aligns with close to each other. The cluster size is controlled by the the island/strait picture: higher density increases the chance that clustering factor (an integer ≥ 2). Implementation-wise, a positive hit occurs or that a nearby negative bucket contains we compute variable-pair frequencies, greedily form and subsumers, extending the last island and triggering larger jumps. merge clusters up to the factor, and then assign low con- secutive indices cluster by cluster (applying the result- 6 Conclusion and Future Work ing translate to the formula). This shortens gaps between We engineered a faithful and fast implementation of CSFLOC, variables that tend to appear together, compresses last- clarifying how the last-1-bit observation can be realized effi- literal positions, and improves the hit rate in the posi- ciently via bucketed, sign-aware data structures, clause learning tive/negative hit scans. The option is enabled by including and variable preordering. Future work includes parallel traversal C in the renaming string and can be combined effectively of counter ranges, richer learning schemes, and broader bench- with , , and (and their compositions). Following Je-R H S marks. belean’s original clustering idea [12], our experiments confirm it is the most consistently helpful preordering References among the options we evaluated. [1] S.Andrei, Counting for Satisfiability by Inverting Resolution, Artificial Intelli- Heuristic notes. Promoting black clauses ( gence Review,Volume 22, Issue 4,339–366, 2004. B ) early tends to [2] E. Birnbaum, E. L. Lozinskii, The Good Old Davis-Putnam Procedure Helps create fewer but longer trailing islands, enabling larger jumps; Counting Models, Journal Of Artificial Intelligence Research, Volume 10, pages many short islands are typically unfavorable, though we do not 457–477, DOI: https://doi.org/10.1613/jair.601, 1999. [3] H. Bennett and S. Sankaranarayanan, Model Counting Using the Inclusion- state formal lemmas due to space. Island- and Horn-aware orders Theory and Applications of Satisfiability Testing - SAT 2011 Exclusion Principle, ( , ) often dominate on over-constrained inputs; randomized , Volume 6695, 362–363, 2011. I H Lecture Notes in Computer Science [4] A. Biere, M. Heule, H. van Maaren, T. Walsh, Handbook of Satisfiability, tie-perturbation inside clusters mitigates adversarial cases. 2nd. edition, , Amsterdam, 2021. IOS Press [5] Cs. Biro and G. Kusper, How to generate weakly nondecisive SAT instances, 5 Empirical Snapshot Proceedings of 11th International IEEE Symposium on Intelligent Systems and Informatics (SISY), 265–269, Subotica, 2013. This section may contain informal terms due to lack of space, but [6] Cs. Biro and G. Kusper, Equivalence of Strongly Connected Graphs and Black- these have been clarified in [13]. and-White 2-SAT Problems, , accepted manuscript, Miskolc Mathematical Notes MMN-2140. The tests were done on iMac macOS Sierra (CP U: 2,5GHz Intel [7] S. A. Cook, The Complexity of Theorem-Proving Procedures, , Proc. of STOC’71 Core i5, Memory: 4GB 1333MHz DDR3). 151–158, 1971. [8] J. K. Fichte, et al., Solving Projected Model Counting by Utilizing Treewidth We follow the measurement settings of our previous report [13]: and its Limits, 314:103810, 2023. Artificial Intelligence the Java implementation of CSFLOC is compared against off-the- [9] M. Davis, G. Logemann, D. Loveland, A Machine Program for Theorem 1 Communications of the ACM shelf CDCL baselines (Glucose 3.0 ) on standard suites (SATLIB Proving, , Volume 5, 394–397, 1962. [10] Carla P. Gomes, Ashish Sabharwal, and Bart Selman, Model Counting, uf/uuf and pigeonhole), generator-based families (WnD UNSAT), Chapter 20 of Handbook of Satisfiability, IOS Press, Amsterdam, 2009. a Black-and-White 2-SAT model of wireless sensor networks [11] K. Iwama, CNF-satisfiability test by counting and polynomial average time, (WSN), and graph-induced SAT encodings (SM/BM/SBB). Time- , Volume 18, Issue 2, 385–391, 1989. SIAM Journal on Computing [12] T. Jebelean and G. Kusper, Multi–Domain Logic and its Applications to outs and machine details are kept fixed across solvers; instances SYNASC’08 SAT, (invited talk), , DOI: 10.1109/SYNASC.2008.93, IEEE Computer are run in a single thread. Society Press, ISBN 978-0-7695-3523-4, 3–8, 2008. [13] G. Kusper, Cs. Biró, Gy. B. Iszály, SAT solving by CSFLOC, the next generation Where CSFLOC shines (and where it does not). CSFLOC is not of full-length clause counting algorithms, Proceedings of IEEE International Conference on Future IoT Technologies 2018, DOI: 10.1109/FIOT.2018.8325589, a random-SAT solver; its strengths appear on over-constrained 2018. or structurally dense inputs where subsumption enables large Solving SAT by an Iterative Version of the [14] G. Kusper, Cs. Biró, jumps. , Inclusion-Exclusion Principle Proceedings of SYNASC 2015, DOI: 10.1109/SYNASC.2015.38, pp. 189–190, 2015. Effect of preordering (clustering and renaming). [15] E. L. Lozinskii, Counting propositional models, Preordering is Information Processing Letters, Volume 41, 327–332, 1992. crucial for CSFLOC because the inner loop scans by last-literal arXiv preprint [16] A. Zaleski, Solving Satisfiability using Inclusion-Exclusion, , DOI: arXiv:1712.06587, 2017. 1 http://www.labri.fr/perso/lsimon/glucose/ 26 Non-redundant Systems of Independence Atoms in Relational Databases ∗ ∗ Lucas Alland Attila Sali Nicole Wu Swarthmore College HUN-REN Alfréd Rényi Institute Harvey Mudd College Swarthmore, PA, United States Budapest, Hungary Claremont, CA, United States Abstract collection of independent atoms over a schema of 𝑛 attributes. Let 𝑅 𝐴 be a relational database schema and 1 = Determine or give good bounds for 𝑓 𝑛 . ( ) { } , 𝐴2, . . . , 𝐴𝑛 𝑟 be a relation over 𝑅. For 𝑋 , 𝑌 𝑅 with 𝑋 𝑌 , 𝑟 is said to ∩ = ∅ ⊂ For the sake of convenience, we identify the schema of 𝑛 satisfy independence atom attributes 𝐴1, 𝐴2, . . . , 𝐴𝑛 {} with the set [𝑛] of the first 𝑛 positive 𝑋 ⊥ 𝑌 , if the projection 𝑟 ( 𝑋 𝑌 ) of 𝑅 to 𝑋 𝑌 is the Cartesian product of the projections 𝑟 𝑋 and 𝑟 𝑌 , ∪ ( ) ( ) Σ integers. It was proven in [2] that every system of independence i.e 𝑟 𝑋 𝑌 𝑟 𝑋 𝑟 𝑌 . Implication of independence atoms is ) = ( ) × ( ) ( atoms has an Armstrong database, that is a database that satisfies an independence atom 𝜎 iff 𝜎. This implies that every non-Σ | = defined naturally and a collection of independence atoms is non- redundant system of independence atoms actually occurs as a redundant, if none of its members is impled by the remaining system satisfied by a particular database, so for us it is enough ones. In the present paper the maximum possible size of non- to consider implications at the schema level. redundant system of independence atoms is investigated. Keywords 2 Lower Bounds One may check easily that 𝑓 1 0 and 𝑓 2 1. An immediate ( ) = ( ) = relational databases, independence atoms, non-redundant system lower bound for 𝑓 𝑛 follows from the observation that none of ( ) 1 the derivation rules introduces new attributes. That is, 𝑋 𝑌 ⊥ Introduction cannot be derived from a set 𝑋 : 𝑖 𝐼 𝑌 if 𝑋 𝑌 Σ = { ⊥ ∈ } ∪ 𝑖 𝑖 𝑖 𝑖 ⊉ In this paper, we investigate an efficient subclass of embedded ∪ ∈ 𝑋 𝑌 for all 𝑖 𝐼. multivalued data dependencies which are called – in accordance Proposition 2.1. 𝑛 3 For ≥ we have with [1] – independence atoms. 𝑛 Definition 1.1. 𝑓 𝑛 . ( ) ≥ A relation 𝑟 satisfies the independence atom 𝑛 ⌈ ⌉ 𝑋 𝑌 between two disjoint sets 𝑋 and 𝑌 of attributes, if for ⊥ 2 all tuples 𝑡 Proof. Let 1 , 𝑡 2 ∈ 𝑟 there is some tuple 𝑡 ∈ 𝑟 which matches the { 𝑛 𝑋 : 2 𝑖 ∪ 𝑌 𝑖 𝑖 1 =} , , . . . , be list of nontrivial 𝑛 ⌈ ⌉ 2 values of 𝑛 𝑡 on all attributes in 1 2 partitions of the 𝑋 and matches the values of 𝑡 on -element subsets of ⌈ ⌉ [ ] Σ = 𝑛 . The system 2 all attributes in 𝑛 𝑌 . 𝑋 { 1 2 𝑖 ⊥ 𝑌 : is clearly non-redundant, as 𝑖 𝑖 = , , . . . , 𝑋 ∪ ⌈ 𝑛 } ⌉ 𝑖 2 In other words, in relations that satisfy 𝑌 𝑖 ⊈ 𝑋 𝑗 ∪ 𝑌𝑗 for 𝑖 ≠ 𝑗. 𝑋 ⊥ 𝑌 , the occurrence □ of 𝑋-values is independent of the occurrence of 𝑌-values. It is interesting to observe that this simple lower bound is If is a collection of independence atoms and 𝜎 is an inde-Σ sharp for small 𝑛. pendence atom, then implies 𝜎, in notation 𝜎, if any Σ Σ | = database relation 𝑟 that satisfies every atom in also satisfies 𝜎. Σ and Proposition 2.2. 𝑓 3 3 𝑓 4 6. ( ) = ( ) = The implication problem was axiomatized by Kontinen , Link and The proof of the first statement is easy, the latter one needs Väänänen [2]. The following four rules are sound and complete detailed case-by-case analysis. system for the implication of independence atoms. We have improved on the simple lower bound above by using (1) 𝑋 (trivial independence, ). a recursive construction. ⊥ ∅ T (2) 𝑋 𝑌 𝑌 𝑋 (symmetry, ). 𝑛 − ⊥ ⇒ ⊥ S1 Theorem 2.3. For 𝑛 ≥ 2 we have 𝑓 ( 𝑛 ) ≥ 𝑓 ( 𝑛 − 1 ) + ⌈ 𝑛 − 1 ⌉ ⊥ (3) 𝑋 𝑌 ∪ 𝑍 ⇒ 𝑋 ⊥ 𝑌 (decomposition, D ). 2 implying ⊥ (4) 𝑋 𝑌 ∧ 𝑋 ∪ 𝑌 ⊥ 𝑍 ⇒ 𝑋 ⊥ 𝑌 ∪ 𝑍 (exchange, E ). That is, 𝜎 iff 𝜎 has a finite derivation from using the rules ∑︁ 𝑖 | Σ 𝑛 𝑛 −1 = Σ − 1 ∑︁ 𝑖 𝑓 𝑛 𝑓 1 . ( ) ≥ ( ) + = above. ⌉ ⌈ ⌉ ⌈ 𝑖 1 𝑖 − A set of independence atoms, is if, for each non-redundant Σ 𝑖 2 2 = 2 𝑖 = 1 atom 𝜎 , 𝜎 𝜎—no atom can be derived from the other ∈ Σ Σ \ { } Proof. The proof is based on the following observation of [2]. ⊭ Σ [ ] ⊥ Let be a collection of independence atoms over 𝑛 and 𝑋 𝑌 atoms. be an independence atom. Let ′ 𝑋 𝑌 Σ = Σ [] = { ∩ ( ∪ ) ⊥ 𝑉 𝑋 𝑌 The following interesting problem was asked by Sebastian 𝑊 𝑋 𝑌 : 𝑉 𝑊 . If there is no non-trivial atom ∩ ( ∪ ) ⊥ ∈ Σ } Link [3]. Let 𝑓 𝑛 denote the largest size of a non-redundant ( ) Σ ∪ = ∪ Σ ̸|= ⊥ 𝑈 𝑉 where 𝑈 𝑉 𝑋 𝑌 , then 𝑋 𝑌 . ⊥ ∈ ′ ∗ Now assume that Σ This research was done under the auspicies of Research Opportunties course at 𝑛 1 − is a non-redundant system of inde- Budapest Semesters in Mathematics pendence atoms of size 𝑓 𝑛 1 over 𝑛 1 . Define 𝑛 ( − ) [ − ] Σ = Σ −1 : 1 with 𝑛 − 1 ∪ { 𝑛 } ⊥ 𝑋 𝑋 ⊂ [ 𝑛 − ] | 𝑋 | = ⌈ 𝑛 . We claim that 2 ⌉ Permission to make digital or hard copies of all or part of this work for personal Σ 𝑛 is non-redundant. Indeed, none of the atoms of the form or classroom use is granted without fee provided that copies are not made or Σ \ {{ } ⊥ } distributed for profit or commercial advantage and that copies bear this notice and 𝑛 { } ⊥ 𝑛 𝑋 can be derived from 𝑛 𝑋 as no other atom the full citation on the first page. Copyrights for third-party components of this contains all attributes of 𝑛 𝑋 . On the other hand, the atoms { } ⊥ work must be honored. For all other uses, contact the owner /author(s). Σ Σ MATCOS-25, October 9th added to 1 𝑛 th − cannot be used in a derivation for an atom in 𝑛−1 and 10 , 2025., Ljubljana, Slovenia since their intersection with 𝑛 1 is a trivial atom of the form [ − ] © 2025 Copyright held by the owner/author(s). ∅ ⊥ 𝑋 . □ 27 MATCOS-25, October 9th th and 10, 2025., Ljubljana, Slovenia Lucas Alland, Attila Sali, and Nicole Wu 3 Upper bounds Proof. Let Σ be non-redundant, suppose we have 𝐾 ⊆ Σ such that 𝐾 𝛼 and then define ⊨ A useful concept in obtaining upper bounds is the concept of atom shapes. 𝐿 = {1 ⊥ 𝑌 ∈ Σ \ 𝐾} Definition 3.1. 𝑆 An atom 𝑋 = { ⊥1 𝑌 is defined as having the shape ∪ 𝑋 ⊥ 𝑌 ∈ Σ \ 𝐾 | 𝑋 ≠ ∅} (|𝑋 |, |𝑌 |). Notice by symmetry (S) that an (𝑎, 𝑏) shape is equiva- 𝑇 = {𝑈 ⊥ 𝑉 ∈ Σ \ 𝐾 | 𝑈 ∪ 𝑉 ⊆ {2, 3, 4, 5}}. lent to an 𝑏, 𝑎 shape. ( ) Σ Then, we may partition by Non-trivial shapes on five attributes, with examples. Σ = ∪ ∪ ∪ 𝐾 𝐿 𝑆 𝑇 . (1) 1, 1 : 1 2 ( ) ⊥ Note, that 𝐿 1 by . Now, recalling the proof that there are | | ≤ D (2) 1, 2 : 1 2 3 ( ) ⊥ no 1, 4 atoms, we can conclude ( ) (3) 2, 2 : 1 2 3 4 ( ) ⊥ (4) 1, 3 : 1 1 2 3 𝑆 𝑇 6. ( ) ⊥ | ∪ | ≤ (5) 1, 4 : 1 2 3 4 5 ( ) ⊥ Σ \ Indeed, since 𝛼 is derived from 𝐾 and 𝑆 𝑇 𝐾, we may use ∪ ⊆ (6) 2, 3 : 1 2 3 4 5 ( ) ⊥ 𝛼 in the derivations from 𝑆 𝑇 . This implies that ∪ Proposition 3.2. 7 𝐾 A maximum sized non-redundant system of | Σ | ≤ | | +. independence atoms over 5 attributes cannot contain an atom of Σ Σ shape Suppose 𝛽 1 2 3 4 5 , and is non-redundant. Note that = ⊥ ∈ (1, 4) . 𝛽 implies 𝛼 by exchange ( ) if 𝛾 1 2 is derivable from 𝛽 . E = ⊥ Σ \ { } Proof. Let 𝛼 1 2 3 4 5 and be a non-redundant set ⊥ Σ = Consider two cases: on 5 attributes, containing (1) If 𝛾 𝛼 . We see that 𝛼 1 ⊥ Σ ⊨ 𝑌 for all ⊭ then the only allowed atoms in Σ \ {𝛽 } are from 𝑌 , 3 2 the sets: ⊆ { , 4 , 5 } . We partition Σ \ { 𝛼 } by the sets 𝑆 and 𝑇 where: 𝑆 1 𝑋 𝑌 𝑋 = { ∪ ⊥ ∈ | ≠ ∅} Σ 𝑃 𝑈 𝑉 𝑈 𝑉 3, 4, 5 , 𝑈 , 𝑉 = { ⊥ | ∪ ⊆ { } ≠ ∅ ≠ ∅} 𝑇 𝑈 𝑉 𝑈 𝑉 2, 3, 4, 5 . = { ⊥ ∈ | ∪ ⊆ { }} Σ 𝑄 2 𝑈 𝑉 𝑈 𝑉 3, 4, 5 , 𝑈 , 𝑉 = { ∪ ⊥ | ∪ ⊆ { } ≠ ∅ ≠ ∅} 𝑅 1 𝑈 𝑉 𝑈 𝑉 3, 4, 5 , 𝑈 , 𝑉 = { ∪ ⊥ | ∪ ⊆ { } ≠ ∅ ≠ ∅} Now define 𝑊 1 2 𝑈 𝑉 𝑈 𝑉 3, 4, 5 , 𝑈 , 𝑉 . = { ∪ ⊥ | ∪ ⊆ { } ≠ ∅ ≠ ∅} ′ = { ⊥ | ∪ ⊥ ∈ } 𝑆 𝑋 𝑌 1 𝑋 𝑌 𝑆 . = ∅ ⊆ { } In particular, 𝐿 , since 𝑌 3, 4, 5 would hold, thus We will show ′ ′ ′ ′ 𝛽 | = Σ 1 ⊥ 𝑌 by D. = 𝑆 ∪ 𝑇 is non-redundant. Take any 𝜎 ∈ Σ Observe that for 𝑈 𝑉 3, 4, 5 , 𝑈 , 𝑉 𝑈 ∪ ⊆ { } ≠ ∅ ≠ ∅ { ⊥ and consider two cases: 𝜎 ′ ′ 𝑈 = ⊥ 𝑉 ∈ 𝑇 Every atom in 𝑆 is derivable from its cor- { 𝑉 , 𝛽 1 2 𝑈 𝑉 by , . So for a given pair 𝑈 𝑉 } | = ∪ ⊥ D E ∪ ⊆ 3, 4, 5 , 𝑈 , 𝑉 at most one atom from the sets } ≠ ∅ ≠ ∅ ′ ∈ ′ Σ responding atom in 𝑆. So if 𝜎 𝑇 were derivable from = { ⊥ ∪ ⊥ Σ ′ ′ 𝜎 \ {} then the same sequence of implications would ′ ′ 𝑉 𝑃 , 𝑄 , 𝑅, 𝑊 can be in . That is, if 𝑄 𝑈 𝑉 : 2 𝑈 ′ ⊨ 𝑉 : 1 2 ∪𝑈 ⊥ 𝑉 ∈ 𝑊 }, then these sets are pairwise disjoint ′ show 𝜎 𝜎 . Σ \ { ′ ∈ 𝑄 }, 𝑅 = {𝑈 ⊥ 𝑉 : 1 ∪ 𝑈 ⊥ 𝑉 ∈ 𝑅} and 𝑊 = {𝑈 ⊥ ′ } 𝜎 𝑋 𝑌 𝑆 Let 𝜎 1 𝑋 𝑌 . Now, we observe: = ⊥ ∈ = ∪ ⊥ ′ ′ ′ and also 𝑃 𝑄 𝑅 𝑊 is a non-redundant system of ∪ ∪ ∪ 𝛼 1 𝑋 𝑌 by ⊥ ∪ D ⊨ independence atoms over the attribute set 3, 4, 5 . Thus, { } | ′ ′ ′ ∪ ∪ ∪ | = | ∪ ∪ ∪| ≤ 𝑃 𝑄 𝑅 𝑊 𝑃 𝑄 𝑅 𝑊 3 implying 1 𝑋 𝑌 𝑋 𝑌 𝜎 by . ⊥ ∪ ∧ ⊥ ⊨ E |Σ| ≤ 4 < 12. So if 𝜎 (2) If 𝛾 then we have an atom from the set Σ ⊨ Σ ′ ′ ′ then with the above two implications \ { } ⊨ 𝜎 we have 𝜎 𝜎. Σ \ { } ⊨ 𝑆 1 𝑋 2 𝑌 𝑋 𝑌 3, 4, 5 = { ∪ ⊥ ∪ | ∪ ⊆ {}} So we conclude is non-redundant. Since 𝑓 4 6, and is a Σ ( ) = Σ ′ ′ set on four attributes, 6. Then, | | ≤ Σ| ≤ Σ With 𝛽 , one such atom will derive 𝛼. Then, 𝐾 2 which | | ≤ ′ forces 9 < 12. | | ′ ′ Σ | = |{ }| + | | + | | = + | | + | | = + | Σ | ≤ 𝛼 𝑆 𝑇 1 𝑆 𝑇 1 7. □ □ Proposition 3.4. 𝑓 (5) ≤ 30 Proof. We use the above two facts to find an upper bound on We note that the previous argument generalizes to all 𝑛 from the given 𝑛 5 case. We can in general conclude that if is a = Σ 𝑓 5 . Let 31 5𝑓 4 1 on 5 attributes. Then, for to be ( ) | Σ | = = ( ) + Σ non-redundant: non-redundant set of maximum size on 𝑛 attributes that contain a 1, 𝑛 1 shape atom: (1) Each attribute from 5 may miss at most 𝑓 4 atoms, ( − ) [ ] ( ) since those atoms must form a non-redundant system on | Σ| ≤ 1 + 𝑓 (𝑛 − 1). the remaining four attributes. Thus, each attribute must appear in at least 𝑓 4 31 6 25 times; | Σ | − ( ) = − = Since our lower bound for 𝑓 𝑛 increases faster than 1 when ( ) 𝑛 3, the maximum-order non-redundant attribute set will ≥ (2) Each atom can contain at most 4 attributes. never contain a 1, 𝑛 1 shape atom. Comparing the possible number of appearances and the neces-( − ) sary number of appearances over all attributes would imply the Proposition 3.3. A maximum sized non-redundant set of in- following inequality: dependence atoms on 5 attributes cannot contain an atom of shape Σ| = (2, 3) . 125 5 24 1 4 124 = ( + ) ≤ | 28 Non-redundant Systems of Independence Atoms in Relational Databases th th MATCOS-25, October 9 and 10, 2025., Ljubljana, Slovenia which clearly does not hold. So for any non-redundant atom-set we obtain Σ, |Σ| ≤ 30. Thus, 𝑓 (5) ≤ 30. □ 2 ( 𝑛 − 1 ) 2 ( 𝑛 − 𝑘 + 1 ) 𝑛 𝑘 ( 𝐿 𝑘 ) = 𝑛 · · . . . · = 2. + 1 1 𝑘 − 1 + 1 𝑘 We can give a general upper bound. Then, the maximum of 𝐿 𝑘 is reached (non-uniquely) when ( ) Proposition 3.5. 𝑓 𝑛 < 3 𝜀 ) ( −) ( 𝑛 𝑘 . = ⌊ ⌋ 2𝑛 3 Proof. Consider the set of all independence atoms to be a Hence, 𝑋 partially ordered set according to the rule 𝑍 𝑊 𝑋 𝑌 if 2 ⊥ ⊥ ≼ 𝑛 ⌊ 2 𝑛 𝑛 ⌋ 𝑛 ( 𝑓 𝑛 ) ≤ 𝐿 ( ⌊ ⌋) = 2 3 < ( 3 − 𝜀 ⊥ 𝑌 ⊨ 𝑍 ⊥ 𝑊 by decomposition ( D ). This is a graded poset 2 𝑛 ) 3 ⌊ 3 ⌋ by the rank function where the final bound can be obtained by applying some asymp- 𝜌 𝑋 𝑌 𝑋 𝑌 . ( ⊥ ) = | ∪ | totics on the binomial coefficient. □ For example, the elements of level 1 are the trivial atoms with the empty set on one side and one attribute on the other. 4 Conclusions A rank 𝑘 element of this poset is covered by 2 𝑛 𝑘 elements We have studied non-redundant systems of independence atoms ( − ) of rank 𝑘 1, since we have two choices to which side of the of relational database schemata. We defined 𝑓 𝑛 to be the largest + ( ) atom the new attribute is added. On the other hand, a rank 𝑘 1 possible size of such a system over a schema of 𝑛 attributes. 𝑓 𝑛 + () element covers 𝑘 1 rank 𝑘 elements. This implies that this graded was determined for small 𝑛 and we gave general lower and upper + poset has the — no antichain is larger than the bounds. We conjecture that Sperner property largest rank level. Since every rank-level is itself an antichain, Conjecture 4.1. 𝑓 5 12 ( ) = . the largest rank-level will be a maximum-sized antichain. Thus, since any non-redundant set must be the subset of some Also, we ask whether the general lower bound construction is antichain, 𝑓 𝑛 is bounded by the size of the largest rank-level: optimal. We believe so, but we do not dare to put it as a conjecture. ( ) 𝑓 𝑛 max 𝐿 𝑘 1 𝑘 𝑛 ( ) ≤ { ( ) | ≤ ≤} References where 𝐿 𝑘 denotes the size of rank-level 𝑘. We consider how ( ) [1] Erich Grädel and Jouko Väänänen. 2013. Dependence and independence. 𝐿 𝑘 relates to 𝐿 𝑘 1 . , 101, 399–410. https://www.jstor.org/stable/23488329. ( ) ( + ) Studia Logica [2] Juha Kontinen, Sebastian Link, and Jouko Väänänen. 2013. Independence in The covering numbers determined above give us the following database relations. In . Leonid Logic, Language, Information, and Computation relation: Libkin, Ulrich Kohlenbach, and Ruy de Queiroz, editors. Springer Berlin 2 𝑛 𝑘 𝐿 𝑘 𝑘 1 𝐿 𝑘 1 . Heidelberg, Berlin, Heidelberg, 179–193. isbn: 978-3-642-39992-3. · ( − ) · ( ) = ( + ) · ( + ) [3] Sebastian Link. 2024. Personal Communication. (2024). Using the above iteratively and 𝐿 1 𝑘 , 𝑘 𝑘 𝑛 2𝑛 ( ) = |{ ⊥ ∅ ∅ ⊥ | ∈ [ ]}| = 29 Scrambler Automaton Block Cipher for IoT Devices Pál Dömösi Géza Horváth University of Debrecen University of Debrecen 4028 Debrecen, Kassai Road 26., Hungary 4028 Debrecen, Kassai Road 26., Hungary domosi@unideb.hu horvath.geza@inf.unideb.hu ABSTRACT to the cipher block by block for encryption. The third part is a scrambler automaton which changes from the state received from In this article, we introduce a new block cipher based on finite the feeder to another state in response to an input signal received automata. Its structure is simple, using few and inexpensive op- from the counter. The new state will be the next output block of erations, making it particularly suitable for lightweight crypto- the cipher, i.e. the next ciphertext block. Decryption is essentially graphic applications. done in the same way, but the role of the scrambler automaton is KEYWORDS taken over by a so-called inverse scrambler automaton. It also changes from the state, the next ciphertext block, received from finite automata, Internet of Things, lightweight cryptography, the feeder to another state in response to an input signal received block cypher from the counter. This response is nothing but the next block of 1 the plaintext (i.e. the decrypted secret text block). The counter INTRODUCTION starts its operation with the same secret initial state as during Communication has undergone significant changes in the 21st encryption. After then sends its current state as input to the century. While communication initially took place between peo- inverse scrambler automaton of the cipher. ple, and then in many cases between computers by the end of The term "counter" is used to refer to a method and apparatus the 20th century, in the 21st century countless small smart de- that is presumed to operate according to a discrete time scale vices communicate with each other and their environment via such that at the start of its operation it is in a fixed state 𝑠0, and the Internet. These devices and this changed environment are in every subsequent time instant 𝑡 its state is an element 𝑠 of 𝑡 collectively referred to as the Internet of Things, or IoT for short. ∈ the nonempty, finite state set 𝑆 , where 𝑠𝑡 𝑆 denotes the state These changes have inevitably forced changes in secret commu- of the counter at the time instant immediately preceding the nication and encryption. Since encrypted communication must → time instant 𝑡 , while 𝑓 : 𝑆 𝑆 is a function adapted to map be implemented using inexpensive and simple tools, procedures the nonempty, finite set 𝑆 to itself in a bijective manner. The that use few simple operations, require little memory and stor- triplet 𝑆 , 𝑠0 S = ( ) , 𝑓 will herein after also be referred to as the age space, and still provide fast and secure communication have base structure of the counter, set 𝑆 will be called the state set of come to the fore. These procedures are collectively referred to as ∈ the counter, state 𝑠0 𝑆 the core of the counter, and function lightweight cryptography. → 𝑓 : 𝑆 𝑆 the state transition function of the counter. A significant step towards lightweight cryptography based In the following, it is assumed that the state transition func- on automata theory was the stream cipher introduced by Pál tions f applied in this application are very simple, preferably Dömösi and Géza Horváth in 2017 [1]. This stream cipher was 128 = { − } ∈ ( ) = + 𝑆 0, 1, . . . , 2 1 , and for each 𝑘 𝑆 , 𝑓 𝑘 𝑘 1, if subjected to thorough testing, which confirmed that the system 128 128 + − ( ) = + = − ) 𝑘 1 < 2 1, and 𝑓 𝑘 0, if 𝑘 1 2 1. is resistant to side-channel attacks, a type of attack that plays a significant role in attacks against IoT devices [2]. The authors 3 TRANSPOSITION-CONTROLLED of paper [1] described a scrambler method in patent [5], demon- strating its use as a pseudo-random number generator in the AUTOMATON articles [3] and [4]. This article describes the use of the scrambler For every 𝑚 1, 2, . . . , define the permutations 𝑃1 =, 𝑃2, . . . , 𝑃𝑚 method described in patent [5] as a block cipher. A well-known over the set { 𝑚 1 , . . . , 2} in the following way. common weakness of symmetric encryption is that a suitable Let 𝑛 be a fixed positive integer power of 2, and let us define method (such as asymmetric encryption) is required to exchange the following permutations that are specified as a product of (synchronize) the secret key without revealing confidential infor- transpositions (for example, for a permutation 𝑃 the transposition mation. Therefore, the cipher presented in this paper does not 9, 13 " which thus denotes such a pair " means that 𝑃 9 13 ( ) ( ) = provide a solution to this difficulty. and 𝑃 13 9 . For specifying these permutations, let us consider ( ) = ) the following algorithm for the vector 1, . . . , 𝑛 : ( ) 2 BASIC CONCEPTS If 𝑛 = 2, then let 𝑃1 = (1, 2), and we are ready. Else, let us consider the vectors 1, . . . , 𝑛 2 and 𝑛 2 1, . . . , 𝑛 , ( / ) ( / + ) In this lecture we consider a novel type of block cipher based on abstract finite automata. This cipher consists of three parts. and let us generate the permutation 𝑃1 such that 𝑃1 is a product of such transpositions where for each 𝑘 1, . . . , 𝑛 2 the first com-∈ { / } One of them is a counter which sends its current state as input ponent of the 𝑘-th factor of this transposition-product is the 𝑘-th to a so-called scrambler automaton, which also belongs to the component of the vector 1, . . . , 𝑛 2 , while the second compo-( / ) cipher. The second part is a feeder, which passes the plaintext nent thereof is the 𝑘-th component of the vector 𝑛 2 1, . . . , 𝑛 . ( / + ) Permission to make digital or hard copies of part or all of this work for personal = ( / + ) ( / + ) ( / ) Taken to an expression: 𝑃1 1, 𝑛 2 1 2, 𝑛 2 2 . . . , 𝑛 2, 𝑛 . or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and If 𝑛 4, then let 𝑃2 = = (1, 2) (3, 4), and we are ready. the full citation on the first page. Copyrights for third-party components of this Else, let us carry out the above process separately for the vec- work must be honored. For all other uses, contact the owner /author(s). ( / ) ( / + ) tor 1, . . . , 𝑛 2 and for the vector 𝑛 2 1, . . . , 𝑛 . The product Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia of the two permutations 1, 𝑛 4 1 2, 𝑛 4 2 𝑛 4, 2𝑛 4 ( / + ) ( / + ) · · · ( / /) © 2024 Copyright held by the owner/author(s). and 2𝑛 4 1, 3𝑛 4 1 2𝑛 4 2, 3𝑛 4 2 3𝑛 4, 4𝑛 4 thus ( / + / + ) ( / + / + ) · · · ( / / ) 30 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Pál Dömösi and Géza Horváth obtained will be the permutation 𝑃 2 = (1, 𝑛/4 + 1) (2, 𝑛/4 + the permutations according to indices 𝑎1, . . . , 𝑎𝑛 ∈ 𝐴, 𝑥1, . . . , 𝑥𝑛 ∈ 2 𝑛 4, 2𝑛 4 2𝑛 4 1, 3𝑛 4 1 2𝑛 4 2, 3𝑛 4 2 3𝑛 4, 𝑋 , 𝛿 𝑎1 𝐵,𝑖 ) · · · ( / / ) ( / + / + ) ( / + / + ) · · · ( / ( (, . . . , 𝑎𝑛), (𝑥1, . . . , 𝑥𝑛 )) = (𝑎”1, . . . , 𝑎”𝑛 ), where 4𝑛 4 . 𝑎”1 𝛿 𝑎1, 𝛿 𝑎 , 𝑥 , 𝑃 / ) = ( ( 𝑖 ( 1 ) 𝑃 𝑖 ( 1 ) ) ) Else, like with the above, let us carry out the process sepa- . . rately for the vectors 1, . . . , 𝑛 4 , 𝑛 4 1, . . . , 2𝑛 4 , 2𝑛 4 / ) ( / + / ) ( / + ( . 1 𝑎” 𝑗 = 𝛿(𝑎𝑗 , 𝛿(𝑎 , 𝑥 , if 𝑗 < 𝑃 𝑗 , and 𝑎” , . . . , 3 𝑛 / 4 ) , ( 3 𝑛 / 4 + 1 , . . . , 4 𝑛 / 4 ) 𝛿 . The product of the four permu-𝑃 (𝑎𝑗, 𝑖 ( 𝑗 ) 𝑃 𝑖 ( 𝑗 ) )) 𝑖 ( ) 𝑗 = tations 𝛿 𝑎” , 𝑥 , if 𝑗 𝑃 𝑗 𝑗 1, . . . , 𝑛 , 𝑖 ( ( 1 , 𝑛 / 8 + 1 ) ( 2 , 𝑛 / 8 + 2 ) · · · ( 𝑛 / 8 , 2 𝑛 / 8 ) , ( 2 𝑛 / 8 + 1 , 3 𝑛 / 8 𝑃 𝑖 ( 𝑗 ) 𝑃 𝑖 ( 𝑗 ) )) ≥ ( ) ( ∈ ) + 1 2𝑛 8 2, 3𝑛 8 2 3𝑛 8, 4𝑛 8 , 4𝑛 8 1, 5𝑛 8 1 4𝑛 8 . ) ( / + / + ) · · · ( / / ) ( / + / + ) ( / + . . 2, 5𝑛 8 2 5𝑛 8, 6𝑛 8 , 6𝑛 8 1, 7𝑛 8 1 6𝑛 8 2, 7𝑛 8 / + ) · · · ( / / ) ( / + / + ) ( / + / + 𝑎” 𝛿 𝑎 , 𝛿 𝑎” . , 𝑥 𝑛 𝑛 𝑃𝑖 ( = ( ( )) 2 7 𝑛 8, 8𝑛 8 thereby obtained will be the permutation 𝑃3 / / ) = ) · · · ( 𝑛 𝑃𝑖 𝑛 ) ( ) For example, if 𝑖 1 and 𝑃1 1, 9 2, 10 3, 11 4, 12 5, 13 = = ( ) ( ) ( ) ( ) () (1, 𝑛/8+1) (2, 𝑛/8+2) · · · (𝑛/8, 2𝑛/8) (2𝑛/8 +1, 3𝑛/8+1) · · · (5𝑛/8, (6, 14) (7, 15) (8, 16), then 6𝑛 8 6𝑛 8 1, 7𝑛 8 1 6𝑛 8 2, 7𝑛 8 2 7𝑛 8, 8𝑛 8 . / ) ( / + / + ) ( / + / + ) · · · ( / / ) 𝑎”1 𝛿 𝑎1, 𝛿 𝑎9, 𝑥9 , because 𝑃1 1 9 and 1 < 𝑃1 1 9 , = ( ( )) ( ) = ( ) ( = ) If 𝑛 16, then let 𝑃4 1, 2 3, 4 . . . , 15, 16 . = = ( ) ( ) ( ) 𝑎”2 𝛿 𝑎2, 𝛿 𝑎10, 𝑥10 , because 𝑃1 2 10 and 2 < 𝑃1 2 10 , = ( ( )) ( ) = ( ) ( = ) Else, like with the above, let us carry out the process sepa- . rately for the vectors 1, , 𝑛 8 , 𝑛 8 1, , 2𝑛 8 , 2𝑛 8 . ( · · · / ) ( / + · · · / ) ( / + . 1, , 3𝑛 8 , 3𝑛 8 1, , 4𝑛 8 , 4𝑛 8 1, , 5𝑛 8 , 5𝑛 8 𝑎”8 · · · / ) ( / + · · · / ) ( / + · · · / ) ( / + = 𝛿 (𝑎8, 𝛿(𝑎16, 𝑥16)), because 𝑃1(8) = 16 and 8 < 𝑃1(8) (= 16), 1, , 6𝑛 8 , 6𝑛 8 1, , 7𝑛 8 , 7𝑛 8 1, , 8𝑛 8 . 𝑎”9 · · · / ) ( / + · · · / ) ( / + · · · / ) = 𝛿 (𝑎9, 𝛿(𝑎”1, 𝑥1)), because 𝑃1(9) = 16 and 9 > 𝑃1(9) (= 16), The product of the eight permutations 1, 𝑛 16 1 2, 𝑛 16 𝑎”10 ( / + ) ( / + = 𝛿 (𝑎10, 𝛿 (𝑎”2, 𝑥2)), because 𝑃1 (10) = 2 and 10 > 𝑃1(10) = 2 𝑛 16, 2𝑛 16 , 2𝑛 16 2, 3𝑛 16 2 3𝑛 16, 4𝑛 16 , 2, ) · · · ( / / ) ( / + / + ) · · · ( / / ) (4𝑛/16 + 1, 5𝑛/16 + 1) (4𝑛/16 + 2, 5𝑛/16 + 2) · · · (5𝑛/16, 6𝑛/16), . . ( . / + / + ) ( / + / + ) · · · ( / / ) 6𝑛 16 1, 7𝑛 16 1 6𝑛 16 2, 7𝑛 16 2 7𝑛 16, 8𝑛 16 , ( 𝑎”16 𝛿 𝑎16, 𝛿 𝑎”8, 𝑥8 , because 𝑃1 16 8 and 16 > 𝑃1 16 = ( ( )) ( ) = () (= / + / + ) ( / + / + ) · · · ( / / ) 8𝑛 16 1, 9𝑛 16 1 8𝑛 16 2, 9𝑛 16 2 9𝑛 16, 10𝑛 16 , 8 . ) (12𝑛/16 + 1, 13𝑛/16 + 1) (12𝑛/16 + 2, 13𝑛/16 + 2) · · · (13𝑛/16, 14𝑛 16 , 14𝑛 16 1, 15𝑛 16 1 14𝑛 16 2, 15𝑛 16 2 ) ( / + / + ) ( / + / +) / For the example included below, let us define the above such · · · ( that the automaton 2 𝑛 B = ( 𝑛𝑙 𝑜𝑔𝑛 𝐴 , 𝑋, 𝛿 is defined such that 15 𝑛 / 16 , 16 𝑛 / 16 ) thus obtained will be the permutation 𝑃 4 = 𝐵 ) ( for any 2 𝑎 , . . . , 𝑎 𝐴, ( + ) ( / + ) · · · ( / / ) ( / + / + ) ) ∈ 1 1 𝑛 1 𝑥 , . . . , 𝑥 𝑋 , the transition , 𝑛 16 1 2, 𝑛 16 2 𝑛 16, 2𝑛 16 2𝑛 16 1, 3𝑛 16 1 𝑛𝑙 𝑜𝑔2𝑛 / ∈ 𝑛𝑙 𝑜𝑔 𝑛 ( 𝛿𝐵 𝑎1, . . . , 𝑎𝑛 , 𝑥1, . . . , 𝑥𝑛𝑙 𝑜𝑔 ( ( ) ( 2𝑛)) is generated by first generat- 2 𝑛 / 16 + 2 , 3 𝑛 / 16 + 2 ) · · · ( 3 𝑛 / 16 , 4 𝑛 / 16 ) ( 4 𝑛 / 16 + 1 , 5 𝑛 / 16 + 1 ) ( ing the state vector that can be obtained by applying the transi- / + / + ) · · · ( / / ) ( / + / + ) 4𝑛 16 2, 5𝑛 16 2 5𝑛 16, 6𝑛 16 6𝑛 16 1, 7𝑛 16 1 ( 6 tion function 𝛿 for the vector 𝑎 , . . . , 𝑎 as a state, and for 1 𝑛 / 16 + 2 , 7 𝑛 / 16 + 2 ) · · · ( 7 𝑛 / 16 , 8 𝑛 / 16 ) ( 8 𝑛 / 16 + 1 , 9 𝑛 / 16 𝐵, 1 ( + 1 𝑛 ) ) ( ( ) 8 the vector 𝑥 1, . . . , 𝑥𝑛 as an input signal. Taken to an expression: 𝑛 / 16 + 2 , 9 𝑛 / 16 + 2 ) · · · ( 9 𝑛 / 16 , 10 𝑛 / 16 ) ( 10 𝑛 / 16 + 1 , 11 𝑛 / 16 + 1 first the transitions 𝛿 𝑎 , . . . , 𝑎 1 𝐵, 1 ( (𝑛), (𝑥1, . . . , 𝑥 ) ( 10 𝑛 / 16 + 2 , 11 𝑛 / 16 + 2 ) · · · ( 11 𝑛 / 16 , 12 𝑛 / 16 ) ( 12 𝑛 / 16 +𝑛 1 , )) are generated. 13 Thereafter, the transition function 𝛿 is applied for the result of 𝐵, 𝑛 / 16 + 1 ) ( 12 𝑛 / 16 + 2 , 13 𝑛 / 16 + 2 ) · · · ( 13 𝑛 / 16 , 14 𝑛 / 16 ) ( 14 𝑛 / 16 + 2 1 this transition and the vector 𝑥 𝑛 + (1, . . . , 𝑥2𝑛) as an input signal. , 15 𝑛 / 16 + 1 ) ( 14 𝑛 / 16 + 2 , 15 𝑛 / 16 + 2 ) · · · ( 15 𝑛 / 16 , 16 𝑛 / 16 ) . If Taken to an expression: the transition 𝛿𝐵, 𝑛 = 32 , the let 𝑃 5 = ( 1 , 2 ) ( 3 , 4 ) · · · ( 31 , 32 )3 (𝛿𝐵,2 (𝛿𝐵,1 ( (𝑎1, . . . , 𝑎 and we are ready.𝑛), Else, by continuing the process in an analogous manner, for (𝑥1, . . . , 𝑥𝑛 )), (𝑥𝑛 +1, . . . , 𝑥2𝑛)), (𝑥2𝑛+1, . . . , 𝑥3𝑛)) is generated every 𝑛 > 32,where 𝑛 is a power of two, we get 𝑃1 1, 𝑛 2 ( / + = (therefore, in the permutation approach the steps of this process 1 2, 𝑛 2 12 𝑛 2, 𝑛 , 𝑃2 1, 𝑛 4 1 2, 𝑛 4 2 𝑛 4, / − ) · · · ( / ) = ( / + ) ( / + ) · · · ( / ) ( have to be implemented applying the above-described permuta- 2 tions). This process is carried on in 𝑙 𝑜𝑔2𝑛 steps, wherein, in the 𝑛 / 4 ) ( 2 𝑛 / 4 + 1 , 3 𝑛 / 4 + 1 ) ( 2 𝑛 / 4 + 2 , 3 𝑛 / 4 + 2 ) · · · ( 3 𝑛 / 4 , 4 𝑛 / 4 ) , 𝑃 3 1, 𝑛 8 1 2, 𝑛 8 2 𝑛 8, 2𝑛 8 2𝑛 8 1, 3𝑛 8 1 2𝑛 8 𝐵,𝑙 𝑜𝑔2𝑛 ( / + ) ( / + ) · · · ( / / ) ( / + / + ( / + = last step, the transition function 𝛿 is applied for the state 2 , 3𝑛 8 2 3𝑛 8, 4𝑛 8 4𝑛 𝑛 ( vector obtained, and for the vector 𝑥 , . . . , 𝑥 as ( / + )) · · · ( / / ) ( ) / + / + ) ( / + / + 8 𝑙 𝑜𝑔 𝑛 1 𝑏𝑙 𝑜𝑔 𝑛 1 , 2 2 − ) 5 𝑛 8 1 4 𝑛 8 2, 5𝑛 8 2 5𝑛 8, 6𝑛 8 6𝑛 8 1, 7𝑛 8 1 6𝑛 8 2, 7𝑛 8 2 7𝑛 8, ) · · · ( / / ) ( / + / + ) ( / + / + ) · · · ( / an input signal. Taken to an expression: the transition 8 𝛿 𝛿 𝛿 𝛿 𝑎1 𝐵,𝑙 𝑜𝑔 𝑛 ( 𝐵,𝑙 𝑜𝑔 𝑛 − 2 2 1 (· · · 𝐵, 2 ( 𝐵, 1 ( (, . . . , 𝑎𝑛 ), (𝑥1, . . . , 𝑥𝑛)), 𝑛 / 8 ) , 𝑃 4 = ( 1 , 𝑛 / 16 + 1 ) ( 2 , 𝑛 / 16 + 2 ) · · · ( 𝑛 / 16 , 2 𝑛 / 16 ) ( 2 𝑛 / 16 + 1 (𝑥 is generated. The , 3 𝑛 / 16 + 1 ) ( 2 𝑛 / 16 + 2 , 3 𝑛 / 16 + 2 ) · · · ( 3 𝑛 / 16 , 4 𝑛 / 16 ) ( 4 𝑛 / 16 𝑛 + 1 , . . . , 𝑥 2 𝑛 )) + . . . , 𝑥 , . . . , 𝑥 𝑛 ( 𝑙 𝑜𝑔𝑛 − 1 )+ 1 𝑛𝑙 𝑜𝑔 2 𝑛 )) 1, 5𝑛 16 1 4𝑛 16 2, 5𝑛 16 2 5𝑛 16, 6𝑛 16 6𝑛 16 + ) ( / + / + ) · · · ( / / ) ( / + / automaton defined in such a manner is a scrambler automaton B 1 having 𝑙𝑜𝑔 2 , 7 𝑛 / 16 + 1 ) ( 6 𝑛 / 16 + 2 , 7 𝑛 / 16 + 2 ) · · · ( 7 𝑛 / 16 , 8 𝑛 / 16 ) ( 8 𝑛 / 16 +𝑛 components that is determined by the automaton 1 A, where the automaton A is the base automaton of the au- , 9 𝑛 / 16 + 1 ) ( 8 𝑛 / 16 + 2 , 9 𝑛 / 16 + 2 ) · · · ( 9 𝑛 / 16 , 10 𝑛 / 16 ) ( 10 𝑛 / 16 + 1, 11𝑛 16 1 10𝑛 16 2, 11𝑛 16 2 11𝑛 16, + ) ( / + / + ) · · · ( / / tomaton . By our definition, is an automaton contolled by B B 12𝑛 16 12𝑛 16 1, 13𝑛 16 1 12𝑛 16 2, 13𝑛 16 2 13𝑛 16, ) ( / + / + ) ( / + / + ) · · · ( / B / transpositions of its state components. So, for short, will be 14𝑛 16 14𝑛 16 1, 15𝑛 16 1 14𝑛 16 2, 15𝑛 16 2 15𝑛 16, ) ( / + / + ) ( / + / + ) · · · ( / / called It is also assumed that transposition-controlled automaton. 16𝑛 16 , 𝑃 1, 3 2, 4 5, 7 6, 8 𝑛 3, 𝑛 1 𝑛 / ) · · · = ( ) ( ) ( ) ( ) · · · ( − − ) ( − 𝑙 𝑜𝑔 𝑛 1 − 2 the transition matrix of the base automaton forms a Latin square. 2, 𝑛 , 𝑃 1, 2 3, 4 𝑛 1, 𝑛 . ) = ( ) ( ) · · · ( − ) 𝑙 𝑜𝑔2𝑛 4 THE NOVEL BLOCK CIPHER For example, if 𝑛 16, then = 𝑃 1 1, 9 2, 10 3, 11 4, 12 5, 13 6, 14 7, 15 8, 16 , The scrambler unit of the cipher is called = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B transposition-cont- 𝑃 2 1, 5 2, 6 3, 7 4, 8 9, 13 10, 14 11, 15 12, 16 , using which, receiving a character string 𝑤 = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) rolled automaton, 𝑃 3 1, 3 2, 4 5, 7 6, 8 9, 11 10, 12 13, 15 14, 16 , having a length of power of two 𝑛 yields a string 𝑔 𝑤 (where = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 𝑃 4 1, 2 3, 4 5, 6 7, 8 9, 10 11, 12 13, 14 15, 16 𝑔 : = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) {0, 1}𝑛 → {0, 1}𝑛 . is a function that bijectively maps the set Let us define the automata 𝑖 𝐴 , 𝑋 , 𝛿 of all possible strings of length 𝑛 onto itself ). 𝐵,𝑖 B 𝑛 𝑛 = ( ) ∈ for each 𝑖 {1, . . . , 𝑙 𝑜𝑔 The function g will hereinafter be referred to as a scrambler 2 𝑛 } such that for any (the definition formula below specifies what was referred to above as the calculation "turns": function. A matrix of which each row and each column is a based on the index 𝑗 a "with comma" or a "without comma" char- permutation of the elements of 𝐻 is a Latin square over the acter component is used in the formula; the formula distinguishes set 𝐻 . An automaton without output is an algebraic structure consisting of two non-empty sets, namely, the state set and the 31 Scrambler Automaton Block Cipher Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia input signal set, and a function named transition function that the character sequence 10331023 is obtained. This character se- maps the Cartesian product of the state set and the input signal quence is the text to be scrambled. Let us assume that the block set onto the state set. Therefore, an automaton without outputs length is eight, i.e., the text to be scrambled constitutes a sin- is usually described in the form 𝐴, 𝑋 , 𝛿 , where 𝐴 is the gle block. Let us assume that we have only one permutation of A = ( ) state set, 𝑋 is the input signal set, and 𝛿 : 𝐴 𝑋 𝐴 is the the form 𝑃1 1, 2 , that is, 𝑛 2. Let the message to be en-× → = ( ) = transition function (which assigns a state to each pair having a crypted be the encoded form of the word "OK", i.e., 10331023. state as its first element and an input signal as its second element). First, the first (two-element) block of the message 10331023, i.e., If for every triplet 𝑎, 𝑏 𝐴, 𝑥 𝑋 , 𝑎 𝑏 implies 𝛿 𝑎, 𝑥 "10" is encrypted. Let us assume that the counter core is 𝑠0 ∈ ∈ ≠ ( ) ≠ = 00, 𝛿 𝑏, 𝑥 then is called a . Obviously, if while the state of the counter is calculated applying the formula ( ) A permutation automaton the transition matrix of forms a Latin square then is a 𝑠𝑡 A A = 𝑠𝑡 −1 +1(𝑚𝑜𝑑 16) (that is, when it attains 16 in the quaternary pemutation automaton. number system, it restarts), and the counter is adapted for pass- Next we show the following statement without proof. (See ing on its states as two-digit quaternary numbers. In this case, Appendix for a proof.) with 𝑎1𝑎2 10, 𝑥1𝑥2 01 𝑎”1 𝛿 𝑎1, 𝛿 𝑎2, 𝑥2 𝛿 1, 𝛿 0, 1 = = = ( ( )) = ( ()) = 𝛿 1, 3 1, 𝑎” 2 𝛿 𝑎2, 𝛿 𝑎”1, 𝑥1 𝛿 0, 𝛿 1, 0 𝛿 0, 2 2, ( ) = = ( ( )) = ( ( )) = ( ) = Theorem 1. A transposition-controlled automaton is a permu- and, denoting 𝑎”1 with 𝑎1, and 𝑎”2 with 𝑎2, considering the next tation automaton if and only if its base automaton is also a permu- counter state 𝑥 1𝑥2 = 02 𝑎”1 = 𝛿 (𝑎1, 𝛿 (𝑎2, 𝑥1)) = 𝛿 (1, 𝛿(2, 0)) = tation automaton. ( 𝛿1, 3) = 1, 𝑎”2 = 𝛿 (𝑎2, 𝛿 (𝑎”1, 𝑥2)) = 𝛿 (2, 𝛿(1, 2)) = 𝛿 (2, 3) = 2, that is, the encrypted block corresponding to the first plaintext We note that, as a consequence of our proposition, plaintext block, i.e., 10, is 12. The first of the two steps is carried out accord- cannot always be decrypted from ciphertext even in the posses- ing to the permutations, with the index of the component xi being sion of the secret keys, if the basic automaton of the automaton identical to the state component applied for the second time (i.e., is not a permutation automaton. Therefore, the cipher is useless not with "itself without comma"). The permutation, however, con- if the basic automaton is not a permutation automaton. tains only one inversion, so it ends with the first combinations. Finally, the randomness of the secret text is helped if the base The second "twist" is already a chain embodiment, wherein the automaton is not only a permutation automaton, but also its transition matrix forms a Latin square. Thus we suppose this index of the input signal character component 𝑥 𝑖 corresponds to the index to be carried over, i.e., to the "itself without comma", property of the discussed cipher. for which the "itself with comma" is calculated. The second block 5 EXAMPLE of the plaintext is 𝑎1𝑎2 = 33, while the subsequent state of the counter is 𝑥 1𝑥2 03.Then 𝑎”1 𝛿 𝑎1, 𝛿 𝑎2, 𝑥2 𝛿 3, 𝛿 3, 3 = = ( ( )) = ( ()) = The encoding and decoding procedure will be presented by the 𝛿 3, 3 3, 𝑎” 2 𝛿 𝑎2, 𝛿 𝑎”1, 𝑥1 𝛿 3, 𝛿 3, 0 𝛿 3, 0 0, ( ) = = ( ( )) = ( ( )) = ( ) = next toy example (where 𝑛 4 only). = and, denoting 𝑎”1 with 𝑎1, and 𝑎”2 with 𝑎2, and considering the 5.1 next counter state 𝑥 1𝑥2 10 (in the quaternary number system, = Encryption 03 1 10) 𝑎”1 + = = 𝛿 (𝑎1, 𝛿(𝑎2, 𝑥1)) = 𝛿 (3, 𝛿 (0, 1)) = 𝛿 (3, 3) = 3, Let us consider the following automaton with four states A = ( ( )) = ( ( )) = ( ) = 𝑎”2 𝛿 𝑎2, 𝛿 𝑎”1, 𝑥2 𝛿 0, 𝛿 3, 0 𝛿 0, 0 1, that is, and four input signals (its transition matrix is shown in Table 1 the encrypted block of the second plaintext block 33 is 31 . The below), which automaton will be called the base automaton of = third block of the plaintext is 𝑎 1𝑎2 10, while the subsequent the exemplary scrambler automaton (in the case of encryption, = = ( ( )) = state of the counter is 𝑥 1𝑥2 11. Then 𝑎”1 𝛿 𝑎1, 𝛿 𝑎2, 𝑥2 the base automaton can be called a "key automaton"): ( ( )) = ( ) = = ( ( )) = ( ( )) = 𝛿 1, 𝛿 0, 1 𝛿 1, 3 1, 𝑎”2 𝛿 𝑎2, 𝛿 𝑎”1, 𝑥1 𝛿 0, 𝛿 1, 1 𝛿 0, 0 1, and, denoting 𝑎”1 with a1, and 𝑎”2 with 𝑎2, consid-( ) = 𝛿 ering the next counter state 𝑥 1 state 0 state 1 state 2 state 3𝑥2 = 12 𝑎”1 = 𝛿 (𝑎1, 𝛿 (𝑎2, 𝑥1)) = input 0: 1 2 3 0 𝛿 (1, 𝛿 (1, 1)) = 𝛿(1, 0) = 2, 𝑎”2 = 𝛿(𝑎2, 𝛿 (𝑎”1, 𝑥2)) = 𝛿 (1, 𝛿(2, 2)) = input 1: ( 3 𝛿 1 0 1 2, 0) = 2, that is, the encrypted block of the third plaintext input 2: 2 3 0 1 block 10 is 22. The fourth block of the plaintext is 𝑎1𝑎2 = 23, input 3: while the subsequent state of the counter is 𝑥 0 1 2 31𝑥2 = 13. Then 𝑎”1 𝛿 𝑎1, 𝛿 𝑎2, 𝑥2 𝛿 2, 𝛿 3, 3 𝛿 2, 3 2, 𝑎”2 𝛿 𝑎2, = ( ( )) = ( ( )) = ( ) = = ( Table 1 𝛿 𝑎”1, 𝑥1 𝛿 3, 𝛿 2, 1 𝛿 3, 1 2, and, denoting 𝑎”1 with ( )) = ( ( )) = ( ) = 𝑎1, and 𝑎”2 with𝑎2, and considering the next counter state 𝑥1𝑥2 = 20 (in the quaternary number system, 13 1 20) 𝑎”1 𝛿 𝑎1, + = = ( In the 0-th row of the transition matrix specified in Table 1 , 𝛿 𝑎2 (, 𝑥1)) = 𝛿 (2, 𝛿(2, 2)) = 𝛿(2, 0) = 3, 𝑎”2 = 𝛿 (𝑎2, 𝛿 (𝑎”1, 𝑥2)) = the states are listed, with the 0-th column thereof containing 𝛿 2, 𝛿 3, 0 𝛿 2, 0 3, that is, the encrypted block of the ( ( )) = ( ) = the possible input signals. The condition that the state set and fourth plaintext block 23 is 33. The ciphertext is therefore input set of the automaton are identical is fulfilled also in this 12312233. As it is shown by this simple example, in this case example; however, in certain embodiments the state and input the basic block size is two (the initial and the output character sets of the automaton may be different. The above transition block, i.e., generally speaking, the first and the second charac- matrix constitutes a . The state set of the automa- ter block " applied in a separate respective step " have the same Latin square ton is 0, 1, 2, 3 , which is identical to the input signal set of the length, as well as the applied input signal block; two characters al- { } automaton, and to the character set of both the plaintext and ready constitute a block), the character sequence to be encrypted the ciphertext. The scrambling and the descrambling operations being processed applying a step length corresponding to this will now be illustrated applying this example. Let us consider basic block size; in this example, the character sequence to be the hexadecimal ASCII code of the word "OK", 4𝐹4𝐵 (the non- encrypted consists of eight characters, so the encryption process encrypted data correspond to the word "OK"). Converting this includes four main steps. As it is illustrated above, a certain type hexadecimal value, 4𝐹4𝐵, into the quaternary number system, of a "double scrambling (mixing)" is applied in each encryption 32 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Pál Dömösi and Géza Horváth step, i.e., the second character block obtained in a given "turn" is decrypting block-encrypted ciphertext data according to the in- recycled into the role of the first character block. vention the scrambler/descrambler apparatus is a scrambler/de- scrambler automaton. The invention also relates to a counter- 5.2 Decryption based full-cycle pseudorandom number generator method and apparatus, in which apparatus a counter passes on its state values Now, let us consider the process of decryption (recovery). The to a scrambler automaton, with the string obtained as a result of ciphertext to be deciphered is therefore the following: 12312233. the scrambling process is represented by the method and appara- The inverse permutation automaton of the base automaton ap- tus either as a single pseudorandom character string, or "dividing plied for scrambling (which base automaton is a permutation this character string into equal-length portions" as a sequence of automaton) is the following (the corresponding transition ma- pseudorandom character strings. trix is illustrated in Table 5; for the process of generating the transition matrix of the inverse permutation automaton from the 6 SUMMARY transition matrix see at Table 2): The aim of this paper is to show a symmetric cipher based on novel technologies. Our solution is an innovation regarding both the idea and the technology, so a new technology based on a −1 state 0 state 1 state 2 state 3 𝛿 new idea is developed. The cipher we have developed can replace input 0: 3 0 1 2 older, out-of-date technology, and due to its simplicity they can be input 1: 1 2 3 0 used well in all cases where older, more complex systems cannot input 2: 2 3 0 1 be used due to memory requirements, operation requirements, input 3: 0 1 2 3 or complexity. Currently, the most common block ciphers use Table 2 several different complex operations, can be attacked by side- channel attacks, whereas the block encryption structure we have developed is much simpler, making the implementation more transparent and resistant to side-channel attacks. We can state in case of our system that it provides high security with simple The first block of the ciphertext is 12 , while the first and operations, so its integration into any application is easy. second generated counter states are 01 and 02. Then 𝑎”2 = 𝛿 −1 −1 −1 −1 𝑎 ( 2 , 𝛿 ( 𝑎 1 , 𝑥 2 )) = 𝛿 ( 2 , 𝛿 ( 1 , 2 )) = 𝛿 ( 2 , 3 ) = 2 , 𝑎 ” 1 = 𝛿 (𝑎1, REFERENCES 𝛿 𝑎”2, 𝑥1 𝛿 1, 𝛿 2, 0 𝛿 1, 3 1, and, denoting 𝑎 [1] Pál Dömösi and Géza Horváth. 2017. A Novel Stream Cipher Based on De-( )) = ( )) = ( ) = ( −1 −1 ′ 1 with terministic Finite Automaton. 𝑎 , and 𝑎 ” with 𝑎 , considering the next counter state 𝑥 𝑥 In: R., Freund; F, Mráz; D, Prusa (eds): Ninth = 1 2 2 1 2 01 𝑎”2 𝛿 𝑎2, 𝛿 𝑎1, 𝑥 1 𝛿 2, 𝛿 1, 0 𝛿 2, 2 0, ( ( )) = ( ( )) = ( ) = = −1 − Workshop on Non-Classical Models of Automata and Applications, (NCMA 2017), 1 − 1 Short Papers. Wien, Austria, Technical University of Vienna. pp. 11–16, 6 p. 𝑎”1 𝛿 𝑎1, 𝛿 𝑎”2, 𝑥2 𝛿 1, 𝛿 0, 1 𝛿 1, 3 1, that is, [2] Pál Dömösi, Géza Horváth, Ferenc Tamás Molnár, Szabolcs Kovács and Adama ( ( )) = ( ( )) = ( ) = = −1 −1 1 Diene. 2021. A side-channel attack against an automata theory based stream the deciphered block of the ciphertext block 12 is 10. The second cipher. SURIKAISEKIKENKYUSHO KOKYUROKU / RIMS KOKYUROKU 2193. block of the ciphertext is 31, while the subsequent two generated pp. 64-72, 9 p. counter states are 03 and 10 −1 Then . 𝑎 ” 2 = 𝛿 (𝑎2, 𝛿 (𝑎1, 𝑥2)) = [3] Pál Dömösi, József Gáll, Géza Horváth, Bertalan Borsos, Norbert Tihanyi and −1 − − 1 Yousef Al Hammadi. 2021. A Full Cycle Length Pseudorandom Number Gener-1 ( ( )) = ( ) = = ( ( )) = 𝛿 1, 𝛿 3, 0 𝛿 1, 0 0, 𝑎” 1 𝛿 𝑎1, 𝛿 𝑎”2, 𝑥1 𝛿 −1 −1 3 ator Based on Compositions of Automata. Informatica–Journal of Computing ( , 𝛿(0, 1)) = 𝛿 (3, 3) = 3, and, denoting 𝑎”1 with 𝑎1, and and Informatics 45 : 2. pp. 179-189, 11 p. 𝑎”2 with 𝑎2, considering the next counter state 𝑥1𝑥2 03 𝑎”2 [4] = = Pál Dömösi, Géza Horváth and Norbert Tihanyi. 2023. Simple chain automaton −1 −1 −1 −1 random number generator for IoT devices. Acta Informatica 60 : 3. pp. 317-329, 𝛿 𝑎2, 𝛿 𝑎1, 𝑥1 𝛿 0, 𝛿 3, 0 𝛿 0, 0 3, 𝑎”1 𝛿 𝑎1, ( ( )) = ( ( )) = ( ) = = ( 13 p. 𝛿 𝑎 2, 𝑥2 𝛿 1 ( ” −1 )) = − ( 3 , 𝛿 ( 3 , 3 )) = 𝛿(3, 3) = 3, that is, the deci- [5] Pál Dömösi and Géza Horváth. 2025. Scrambler Apparatus and Method in Particular for Cryptographic Applications, and Descrambler Apparatus and phered block of the second ciphertext block 23 is 33. The third Method Therefor. Patent No. US 12,328,384 B2, Filing Year: 2025, USA Patent. block of the ciphertext is 22 , while the subsequent two generated Filing Number: 17/908,301. counter states are 11 and 12. Then 𝑎” 2 𝛿 𝑎2, 𝛿 𝑎1, 𝑥2 ( ()) = = −1 𝛿 −1 −1 −1 2 ( , 𝛿 ( 2 , 2 )) = 𝛿 ( 2 , 0 ) = 1 , 𝑎 ” 1 = 𝛿 (𝑎1, 𝛿(𝑎”2, 𝑥1)) = −1 −1 ( ( )) =( ) = 𝛿 2, 𝛿 1, 2 𝛿 2, 0 1, and, denoting 𝑎”1 with 𝑎1, and 𝑎”2 with 𝑎2, considering the next counter state 𝑥1𝑥2 11 𝑎”2 = = −1 − − 11 −1 ( ( )) = ( ( )) = ( ) = = ( 𝛿 𝑎2, 𝛿 𝑎1, 𝑥1 𝛿 1, 𝛿 1, 1 𝛿 1, 0 0, 𝑎”1 𝛿 𝑎1, 𝛿 𝑎”2, 𝑥2 𝛿 1, 𝛿 0, 1 𝛿 1, 3 1, that is, the deci-)) = ( ( )) = ( ) = ( −1 −1 phered block of the third ciphertext block 12 is 10. The fourth block of the ciphertext is 33, while the subsequent two generated counter states are 13 and 20 −1 Then . 𝑎 ” 2 = 𝛿 (𝑎2, 𝛿 (𝑎1, 𝑥2)) = −1 −1 −1 ( ( )) = ( ) = = ( ( )) = 𝛿 3, 𝛿 3, 0 𝛿 3, 0 2, 𝑎” 1 𝛿 𝑎1, 𝛿 𝑎”2, 𝑥1 𝛿 −1 −1 3 ( , 𝛿 ( 2 , 2 )) = 𝛿(3, 0) = 2, and, denoting 𝑎”1 with 𝑎1, and 𝑎”2 with 𝑎2, considering the next counter state 𝑥1𝑥2 13 𝑎”2 = = 𝛿 −1 −1 −1 −1 𝑎 ( 2 , 𝛿 ( 𝑎 1 , 𝑥 1 )) = 𝛿 ( 2 , 𝛿 ( 2 , 1 )) = 𝛿 ( 2 , 1 ) = 3 , 𝑎 ” 1 = 𝛿 (𝑎1, 𝛿 𝑎 3 3 2 ( ” −1 − , 𝑥 2 )) = 𝛿 ( 2 , 𝛿 ( , )) =1 𝛿 (2, 3) = 2, that is, the deci- phered block of the fourth ciphertext block 22 is 23. Combining the deciphered blocks, the plaintext 10331023 is obtained. Con- verting this back into hexadecimal form, the ASCII-coded version of the text "OK" is obtained. In the case of both the cryptographic apparatus for block- encrypting plaintext data and the cryptographic apparatus for 33 Scrambler Automaton Block Cipher Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia 7 APPENDIX 𝑛 there exists a 𝑗 ∈ { 1 , . . . , 𝑛 } with 𝑎 . Let 𝑗 ≠ 𝑏 𝑗 ( 𝑥 1 , . . . , 𝑥 𝑛 ) ∈ 𝑋 Let be an arbitrary input letter of 𝑖 . Put 𝑎 , . . . , 𝑎 𝑛 be an arbitrary positive integer and let A 𝑖 = ( 𝐴, 𝑋 𝑖 , 𝛿 𝑖 ) , 𝑖 = 1𝑛 ) B ′′ ′′ ( 1 𝛿 ′′ ′′ 𝑎 ( ( 𝑗 1 , . . . , 𝑎 𝑛 ) , ( 𝑥 1 , . . . , 𝑥 𝑛 )) and ( 𝑏 , . . . , 𝑏 , . . . , 𝑛 be a finite sequence of automata with a common state set 1 𝑛) = 𝐴. Define the automaton 𝐴, 𝑋1 𝑋2 𝑋𝑛, 𝛿 such that for 𝑗 B = ( × ×· · ·× ) 𝛿 ( (𝑏1, . . . , 𝑏𝑛), (𝑥1, . . . , 𝑥𝑛)). every Suppose 𝑗 < 𝑃 𝑖 𝑗 . Then, by definition, 𝑎”𝑗 𝛿 𝑎𝑗, 𝛿 𝑎 , ( ) = ( ( 𝑎 ∈ 𝐴, ( 𝑥 1 , . . . , 𝑥 𝑛 ) ∈ 𝑋 1 × 𝑋 2 ×· · ·× 𝑋 , 𝛿 ( 𝑎, ( 𝑥 1 , . . . , 𝑥 𝑃 𝑖 ( 𝑗 ) 𝑛 𝑛 )) = 𝛿𝑛 𝛿𝑛 − (1(· · · 𝛿1(𝑎, 𝑥1), 𝑥2) 𝑥 and 𝑏” 𝛿 𝑏 , 𝛿 , 𝑃 ( ( · · ·𝑎 , 𝑥 , 𝑥 𝑖 ( 𝑗 ) )) 𝑗 = 𝑗 . Therefore, by our as-𝑃 𝑖 ( 𝑗 ) 𝑃 𝑖 ( 𝑗 ) )) 𝑛 − 1 ) , 𝑥 𝑛 ) . Then we say that B is ′′ ′′ ′′ ′′ ′′ ′′ sumptions, 𝑎 𝑎 , . . . , 𝑎 ≠ 𝑗 𝑏 . Therefore, 𝑏 the temporal product of A 1 , . . . , A 𝑛 . Then we also say that A 𝑖 = 𝑗 ( , . . . , 𝑏 . 1 𝑛 ) ≠ ( 1 𝑛 ) ( Suppose 𝑗 𝐴, 𝑋 𝑖 , 𝛿 𝑖 ) , 𝑖 = 1 , . . . , 𝑛 are component-automata of the temporal ≥ 𝑃𝑖 ( 𝑗 ). Then there are two possibilities. First, product 𝑎 𝑏 ” ” ′′ ′′ ′′ ′′ B . 𝑃 𝑖 ( 𝑗 ) ≠ . In this case, 𝑎 , . . . , 𝑎 𝑃 ) ≠ 𝑏) again. 𝑖 ( 𝑗 ) ( 1 𝑛 ( , . . . , 𝑏 1 𝑛 Proposition 2. A temporal product of automata is a permuta- Second, 𝑎” 𝑏” . In this case, 𝑎” 𝛿 𝑗 𝑃 ( 𝑖 𝑗 ) = 𝑃 𝑖 ( 𝑗 ) = (𝑎𝑗, 𝛿 (𝑎” , 𝑃 𝑖 ( 𝑗 ) tion automaton if and only if each of its component automata is a )) 𝑥 , and 𝑏” 𝛿 𝑏 , 𝛿 𝑏 ” 𝛿 𝑗 𝑃 𝑖 ( 𝑗 )) = ( 𝑗 ( , 𝑥 𝑃 𝑖 ( 𝑗 ) 𝑃 𝑖 ( 𝑗 ) )) =(𝑏 𝑗, 𝛿 (𝑎” , 𝑃 𝑖 ( 𝑗 ) 𝑥 . Because the base automaton is a permutation automaton, ( ) permutation automaton. 𝑃 𝑗 we obtain ′′ ′′ ′′ ′′ ′′ ′′ 𝑎 , . . . , 𝑎 ≠ 𝑖 𝑗 𝑗 1 𝑛 1 𝑛 ( ) ≠ ( ) 𝑏 which leads to , . . . , 𝑏 𝑎 𝑏 Now we are ready to prove our main statement. again. Therefore, if the base automaton is a permutation au- Theorem 3. A transposition-controlled automaton is a permu- tomaton then the transposition controlled automaton is also a tation automaton if and only if its base automaton is also a permu- permutation automaton. tation automaton. Next we assume that the base automaton is not a permutation automaton. Then there are 𝑎, 𝑏 𝐴, 𝑥 𝑋 with 𝑎 𝑏 and ∈ ∈ ≠ Proof: Consider a transposition-controlled automaton B = 𝛿 (𝑎, 𝑥) = 𝛿 (𝑏, 𝑥). Consider a pair (𝑎1, . . . , 𝑎𝑗 −1, 𝑎 𝑗, 𝑎 𝑗+1, . . . , 𝑎𝑛), 1 𝑗 𝑗 1 𝑛 , 𝑏 , 𝑎 , . . . , 𝑎 . ( 𝑛 𝑛 𝐴 , 𝑋, 𝛿 , 𝑖 1, . . . , 𝑙 𝑜𝑔 is a tem-𝐵,𝑖 ) ∈ { 2 𝑛 } . Obviously, then B ′′ Assume 𝑗 < 𝑃 𝑖 ( 𝑗 ) . Then choosing 𝑥 𝑃 ( 𝑖 ( 𝑛 𝑛𝑙 𝑜𝑔2𝑛 𝐴 , 𝑋, 𝛿 consisting of component-automata 𝐵 ) B𝑖 = (𝑎1, . . . , 𝑎 𝑗 − + )) poral product of components , 𝑖 𝑖 B ∈ {1, . . . , 𝑙 𝑜𝑔2𝑛} . Hence, by ′′ 𝑏. Finally, assume 𝑗 𝑃 𝑗 ≥ 𝑗 𝑗 𝑗 ) = 𝑥 , we obtain 𝑎 = 𝑖 𝑖 ( ) ( ) ≠ 𝑗 we . Then because of 𝑃 𝑗 Proposition 2, is a permutation automaton, if and only if, each B have 𝑎 𝑏 and thus 𝑎 𝑏 . But then 𝑎” 𝑗 𝑃 ( 𝑗 ) = ′′ ′′ = = of its component-automata 𝑃 𝑗 ( B 𝑖 , 𝑖 1 𝑛 are permutation 𝑃 , . . . , 𝑙 𝑜𝑔 2 ) ∈ { } (𝑗 ) 𝑃 (𝑗 ) automata. 𝛿 𝑎 𝑗 , 𝛿 𝑎” , 𝑥 𝑏 ” 𝛿 𝑏 , 𝛿 𝑏 , 𝑥 𝑃 ( 𝑖 𝑗 ) 𝑃 ( ( ′′ ( 𝑖 𝑖 𝑖 𝑗 𝑗 ) = ( ( )) 𝑗 = 𝑗 𝑃 𝑗 𝑃 𝑗 ( ) () )) = ” 𝑏 again. First we assume that the base automaton is a permutation Thus, if the base automaton is not a permutation automa- automaton. Consider a positive integer 𝑗 1, . . . , 𝑛 and a pair 𝑎1 ton then the transposition controlled automaton is also not a ∈ { } ( , . . . , 𝑎𝑛 , 𝑏1, . . . , 𝑏𝑛 𝐴 with ) ( 𝑛 permutation automaton. The proof is complete. ) ∈ ( 𝑎 1 , . . . , 𝑎 𝑛 ) ≠ ( 𝑏 1 , . . . , 𝑏 𝑛 ) . Then 34 Automata for context-free trace languages and permutation languages (extended abstract) Benedek Nagy Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University Famagusta, North Cyprus, via Mersin 10, Turkey and Department of Computer Science, Institute of Mathematics and Informatics Eszterházy Károly Catholic University, Eger, Hungary nbenedek.inf@gmail.com ABSTRACT and linear classes [18, 25, 26], the class of metalinear lan- Traces are sets of equivalent words (sequences of actions); guages [5] is properly between the linear and context-free they are efficient tools to describe parallel processes. A trace classes and the class of permutation languages [9, 13, 14] language is a language of traces usually used in linearized is properly between the context-free and context-sensitive form, i.e., it is a standard formal language that contains all classes. In this paper, we consider the class of permutation words representing the traces of the language. Trace lan- languages (they are formally recalled in Section 2, from [14]). guages can be described by a base language and with a(n in)dependency relation on the alphabet. Context-free trace One of the main motivations for studying permutation gram- languages are based on context-free languages. Permutation mars/languages comes from computational linguistics, as grammars are context-free grammars extended by permuta- there are various non-context-free structures that occur in tion rules (type AB → BA productions). They generate natural languages, in general. The non-context-free lan- permutation languages. In this paper an extended variant n n n guages of multiple agreements { a b c}, cross dependen- of the pushdown automata is shown that is able to charac- n m n m ∗ cies { a b c d } and copy { ww | w ∈ { a, b }} are structures terize the permutation languages. We also show that every that occur in some natural languages. There is a big gap context-free trace language is a permutation language, and between the efficiency of context-free and context-sensitive thus, all of these languages are also accepted by such au- grammars. Although, the class of context-free languages has tomata. some nice computational properties, they are not enough to describe several phenomena of the world. On the other Categories and Subject Descriptors hand, the context-sensitive family is too large, and has many inconvenient properties (e.g., PSPACE-complete word prob- F.4.3 [Formal Languages] lem), thus, generally, in applications it is not used. Several branches of extensions of context-free grammars were intro- General Terms duced by controlling the derivations in various ways [2]. In Theory, Automata, Languages, Parallel processes this paper, we do not use any additional control, we focus on permutation grammars that allow to use context-free pro- Keywords ductions and permutation (aka. interchange) rules (of type AB → BA). These additional rules are monotone rules hav- Permutation grammars, formal languages, traces and trace ing exactly the same letters in both sides. Permutation lan- languages, modeling parallel processes, pushdown automata guages were analyzed from computational linguistical point of view in [15, 19] showing, e.g., that each of the three above 1. INTRODUCTION mentioned famous mildly context-sensitive languages can be The Chomsky type grammars and the generated language obtained as an intersection of a permutation language with families are very basic and important parts of theoretical a regular language. The word problem was addressed in [4, computer science [5, 29]. There are various refinements of 20, 21] proving that, in general, it is NP-complete. Hier- the hierarchy where some further classes are properly in- archy, based on the lengths of the permutation rules (e.g., serted: e.g., the class of union-free regular languages is a ABC → CBA has a length 3) was established in [16] and subregular class [1, 12], the class accepted by deterministic an infinite hierarchy was presented in [8]. These grammars 2-head automata (2detLIN) is properly between the regular and languages are also connected to concurrency and the- ory of parallel processes, where the order of some processes can be interchanged (or may go in parallel). The relation semi-commutation is known as a mapping that allows to permute two consecutive letters of a word if their order was in the given one. A partial commutation allows to permute two consecutive letters independently of their original or- der [28]. They are intensively studied by the connection to Petri-nets. Traces and trajectories [10] give formal frame- work to describe these processes [3]. Commutative closure 35 allows all permutations. In Section 3, we formally recall and is a non-context-free language, and its intersection with the describe some of these concepts. ∗ ∗ ∗ n n n regular language a b c results in the language { a b c}. The structure of this paper is as follows. In the next section we recall formal definitions for permutation languages. In There is a normal form for these grammars [14]: Every per- Section 3, we show that all context-free trace languages are mutation language can be generated by a grammar where permutation languages. Further, in Section 4, we present a every production is in one of the following forms: AB → new variant of pushdown automata that is able to recognize BA, A → BC, A → B, A → a where A, B, C ∈ N and a ∈ T . permutation languages. Because the lack of space, for some It is still open whether the chain rules A → B can be elim- of our results, instead of their formal proofs only their proof inated. Note, that in fact, these productions are in Kuroda ideas are shown. normal form of context-sensitive grammars, moreover each non-context-free production is of the form AB → BA. 2. PERMUTATION GRAMMARS Here, we recall some important definitions. We assume that 2. Let a permutation grammar G be given Definition the reader already knows the basic concepts of formal lan- generating L. The language L b generated by the context-free guages, generative grammars and automata theory. For any grammar obtained from G by deleting its non-context-free non-explained concept the reader is referred, e.g., to [5, 29]. productions is a basis language of L. We denote the length of a word u by |u| and the empty word by λ. However, in this paper, we do not care if λ is in a language or not, that is, we consider two languages (or The basis language is usually not uniquely defined for a formalisms defining the languages) equivalent if they may permutation language, it depends on the used grammar. differ at most in the empty word. However, any basis language L b is letter equivalent to L (in Parikh sense). This fact has the following two impor- A generative grammar is a tuple G = (N, T, S, P ) with non- tant consequences: For each basis language Lb ⊆ L. Fur-terminal and terminal alphabets, N and T , respectively, ther, all permutation languages are semi-linear (in Parikh start symbol S ∈ N and the finite set P of productions (aka. sense). This latter fact can be used, e.g., to show that some derivation rules). Each production is of the form p u → v context-sensitive languages (e.g., { a| p is a prime}) are not with ∗ ∗ ∗ u ∈ ( N ∪ T ) N ( N ∪ T ) and v ∈ ( N ∪ T ). A gram- permutation languages. Other such tools are the so-called mar is context-free if u ∈ N for each of its productions. A interchange lemmas proven in [14, 16]. grammar is monotone if |u| ≤ |v| for each productions. A language is context-free (context-sensitive) if it is generated 3. CONTEXT-FREE TRACE LANGUAGES by a context-free (monotone, resp.) grammar. This section starts by describing some formal concepts desc- It was known already in the early 70’s that every monotone ribing/modeling parallel computations. The commutative grammar is equivalent to a grammar having only produc- closure of a language L is the set of all words having Parikh- tions of the following types AB → AC, AB → BA, A → vectors included in the Parikh-map of the language, i.e. BC, A → B and A → a (where A, B, C ∈ N and a ∈ T ). for each word of the language all words that are permu- On the one hand, in 1974 Penttonen showed that one-side tations of its letters are included. The commutative closure context-sensitivity is enough to obtain the whole context- n n n of { a b c} is the language Lp that is shown in Example 1. sensitive language class [27], i.e., grammars with only rules of type AB → AC, A → BC, A → B, A → a suffice. On closure of itself, e.g., A language is called commutative if it is the commutative Lp is commutative. Now some further the other hand, in Turing-machine simulations the rules of concepts are recalled formally based on [3, 6, 11]. type AB → BA can frequently be used representing the movement of the head of the machine. Furthermore, the grammars having non-context-free rules only in the form 3. Let T be a finite alphabet. A reflexive and Definition AB → BA generate the class of permutation languages, a symmetric binary relation D ⊂ T × T is a dependency rela- proper subclass of the context-sensitive class. tion on T : (a, a) ∈ D for every a ∈ T , further, if (a, b) ∈ D Now we define formally the grammar and language class we for some letters a, b ∈ T , then also (b, a) ∈ D. The indepen-are working with. dency relation (or also called commutation) induced by D is I = (T × T ) \ D. Obviously, I is irreflexive and symmetric. Let 0 0 ∗ w ≡ w if w = uabv , w = ubav and a, b ∈ T , u, v ∈ T, Definition 1. A grammar G = (N, T, S, P ) is a permu- ∗ ( a, b ) ∈ I . The equivalence relation ≡ induced by I on T tation grammar if P contains context-free rules and only ∗ is called a partial commutation . For each word w ∈ T the special type of non-context-free rules, namely, interchange set of words that are equivalent to w form the commutation (aka. permutation) rules, which are in the form AB → BA class [w] of w. These classes are called traces. The defini- (A, B ∈ N). We denote the class of permutation languages, tion is also extended to languages: [L] = {[w] | w ∈ L}, then the languages that are generated in this way, by PERM. [L] is a trace language that consists of traces. Further, it is usual to use the language 0 S 0 0 L = [ w ] = { w | [ w] ∈ [L]}. [w]∈[L] Example 1. Let Gp = ({S, A, B, C}, {a, b, c}, S, Pp) be a This language is the linearization of the trace language [L], permutation grammar with Pp = {S → ABC, S → ASBC, or shortly (if it is not confusing) also called a trace lan- AB → BA, AC → CA, BC → CB, CB → BC, A → a, guage. The equivalence class of the empty word is called B → b, C → c}. The language L p containing all words empty trace. Let L be a context-free (regular) language, and with the same number of a-s, b-s and c-s is generated. This I be an independency relation. Further, let [M ] be the set 36 of classes induced by the independency relation I among the By the help of the temporary storage the PDATS can simu- words over the alphabet T , i.e., the trace languages defined late arbitrary context-free derivations. Since it has no inner by I. Then [L] ⊂ [M ] is a context-free (rational, resp.) memory (it has no states) it cannot do more. By inductions trace language. Further, CFTRACE and RAT denote the on the set of configurations of the automata and on the set sets of (linearizations of ) the context-free and rational trace of sentential forms of the grammar in normal form it can be languages, respectively. proven that we have: Trajectories of grid paths are closely related since by per- 1. The class of languages accepted by push-Proposition muting the order of some steps leads to the same target point down automata with temporary storage coincides with the in the grid. Trajectories, traces and trace languages are in- class of context-free languages. tensively studied in relation to modeling parallel processes and concurrency theory [7, 10, 22]. Definition 5. A pushdown automaton with binary per- Theorem 1. mutation (PDABP) is a tuple (T, Γ, δ, ρ, Z0). The compo- CFTRACE is a subset of PERM . nents T, Γ , δ, Z 0 are the same as at PDA, and ρ is a binary relation on Γ × Γ. A configuration of a PDABP is a pair Proof. Let L be a context-free language and I be an in- (v, z) (as at a PDA). The relation ` on configurations is dependency relation on the alphabet T . Let G = (N, T, S, P ) defined as follows. Suppose λ ∈ δ(a, A) with a ∈ T, A ∈ Γ, be a grammar generating the context-free language L in a ∗ ∗ then for all v ∈ T and z ∈ Γ: (av, zA) ` (v, z). Suppose 0 0 0 Chomsky normal form. Let G = ( N , T, S, P ) be a new 0 ∗ ∗ z ∈ δ ( λ, A ) with A ∈ Γ , then for all v ∈ T and z ∈ Γ : 0 grammar with N = N ∪ { X a | a ∈ T } , where each X a is a 0 ( v, zA ) ` ( v, zz ) . Suppose ( A, B ) ∈ ρ with some A, B ∈ Γ new nonterminal. Let 0 ∗ P = P ∪ { X a → a | a ∈ T } ∪ { A → then ( v, zBA ) ` ( v, zAB ) . Further, ` is defined as usual. Xa | A → a ∈ P, A ∈ N} \ {A → a | A ∈ N, a ∈ T } ∗ ∗ Finally, L = { v | ( v, Z 0 ) ` ( λ, λ ) , v ∈ T} is the language 0 0 00 Clearly, G also generates L . Now, let G t = ( N , T, S, P ) accepted by the PDABP. with 00 0 P = P ∪ {X X → X X | (a, b) ∈ I}. The permuta- a b b a tion grammar 00 G generates the context-free trace language Since the new operation, the binary permutation can inter- defined by L and I. change the top two symbols in the stack, PDABP can sim- ulate left-most derivations in permutation grammars. The next statement is the consequence of the fact that left-most Each commutative semi-linear language is in RAT, and also derivations generate only context-free languages indepen- in CFTRACE, thus, it is in PERM. dently of the form of the productions of the grammar [29]. 4. Note that left-most derivations without loosing the genera- NEW EXTENDED VARIANTS OF tive power in the context-sensitive case are shown in [17]. PUSHDOWN AUTOMATA In this section, we define some extensions of the pushdown automata. Note that here we use only stateless pushdown 2. The class of languages accepted by Proposition automata which are usually used without loss of general- PDABP coincides with the class of context-free languages. ity [29]. Moreover, we use automata with special transition mappings that are closely related to grammars in normal forms. Formally, a 6. A pushdown automaton with temporary pushdown automaton (PDA) is a tuple Definition ( storage and binary permutation (PDATSBP) is a pentuple T, Γ , δ, Z 0 ) with the input and stack alphabets T and Γ, respectively, (T, Γ, δ, ρ, Z0 ). The components are the same as at PDABP. Z 0 ∈ Γ is the initial symbol in the stack, and δ Γ∗ : ( T ∪ { λ} ) × Γ → 2 is the transition relation where only The configuration is defined in the same way as at PDATS. finite subsets of Γ∗ The relation ` of a PDATS is extended with the following are used. A PDA simulates left-most derivations in a corresponding context-free grammar. steps. Suppose (A, B) ∈ ρ then (v, zBA, t) ` (v, zAB, t) (with any ∗ ∗ ∗ v ∈ T , z, t ∈ Γ ). The definitions of ` and of the accepted language are straightforward. Definition 4. A pushdown automaton with a tempo- rary storage (PDATS) is a 4-tuple (T, Γ, δ, Z 0) as the push- down automata. The stack alphabet Γ is also used as the Actually, in PDATSBP we combined PDATS and PDABP. set of possible symbols in the temporary storage. A con- They can be redefined such that the top of the stack and figuration of a PDATS is a triple (v, z, t), where v is the the top of the temporary storage are interchanged. Although remaining input, z is the contents of the stack and t is both PDATS and PDABP are equivalent to PDA concerning the contents of the temporary storage. The computation their accepting capacity, we have the next theorem. step relation (`) is defined among the configurations as fol- lows. Suppose λ ∈ δ(a, A) with a ∈ T, A ∈ Γ, then for all ∗ ∗ v ∈ T and z ∈ Γ: (av, zA, λ) ` (v, z, λ). Suppose 2. The class PERM is accepted by PDATSBP. Theorem z 0 ∗ ∗ ∈ δ ( λ, A ) with A ∈ Γ , then for all v ∈ T and z, t ∈ Γ: ( 0 ∗ 0 ∗ v, zA, t ) ` ( v, zz , t ) . Moreover for all v ∈ T and z, z ∈ Γ The proof goes by induction. There is a correspondence be- ( 0 0 0 0 v, zA, z ) ` ( v, z, z A ) and ( v, z, z A ) ` ( v, zA, z). We tween the configurations of the automata and the sentential can also define ∗ ` as the reflexive and transitive closure of forms of the grammars (using derivations where it is signed `. By this relation we can define the accepted language: which part of the sentential form is rewritten by an applica- L ∗ ∗ = { v | ( v, Z , λ ) ` ( λ, λ, λ ) , v ∈ T}. ble production). 0 37 Now, our example is continued. [13] B. Nagy. Languages generated by context-free and type AB → BA rules. In: Proc. CINTI 2007: 8th Int. Example 1 (cont.) The language Lp is accepted by the Symp. of Hung. Research. Comput. Intelligence Inf., PDATSBP M = ({a, b, c}, {S, A, B, C}, δ, {(A, C), (A, B), Budapest, Hungary, pp. 563–572, 2007. (B, C), (C, B) }, S), where CBA ∈ δ(λ, S), CBSA ∈ δ(λ, S), [14] B. Nagy. Languages generated by context-free and λ ∈ δ(a, A), λ ∈ δ(b, B), λ ∈ δ(c, C). type AB → BA rules. J Autom. Lang. Combin. 14 pp. 175–186, 2009. As a corollary of Theorems 1 and 2, we have the following [15] B. Nagy. Permutation languages in formal linguistics. important result. In: Proc. IWANN 2009, Part I, LNCS 5517 pp. 504—511, 2009. Corollary 1. There is a PDATSBP for every context- [16] B. Nagy. On a hierarchy of permutation languages. In: Automata, Formal Languages and Algebraic Systems, free trace language that accepts it. pp. 163–178, World Scientific, Singapore, 2010. 5. CONCLUSIONS Grammars. In: [17] B. Nagy. Derivation Trees for Context-Sensitive Automata, Formal Languages and Permutation grammars and the newly investigated variant of Algebraic Systems, pp. 179–199, World Scientific, pushdown automata are capable to model parallel processes Singapore, 2010. where the base language is context-free and also some lin- 0 0 [18] B. Nagy. 5 → 3 Sensing Watson-Crick Finite guistically important structures. It is known that the word Automata. In: G. Fung, ed.: Sequence and Genome problem for permutation grammars is NP-complete in gen- Analysis II - Methods and Applications, iConcept eral [21]. It is a future task to analyse what is the case for Press, pp. 39—56, 2010. context-free trace languages or at least for those of them [19] B. Nagy. Linguistic power of permutation languages which may appear in modeling and in applications. Re- by regular help. In: Bio-inspired models for natural lation of these classes to languages accepted by pushdown and formal languages. pp. 135–152, Cambridge automata with translucent letters [23, 24] should also be Scholars, 2011. studied in the future. About the normal form for permu- [20] B. Nagy. On the NP-completeness of the word tation grammars, it is an open problem whether the chain rules A → problem for permutation grammars. In: Abstract B can be eliminated. volume of the Int. Workshop in honor of Masami Ito’s 6. REFERENCES Satellite Workshop in Kyoto 77th birthday and P´ al D¨ om¨ osi’s 75th birthday: DLT’s , Japan, September 2018. [1] S. Crvenkovi´ c, I. Dolinka and Z. ´ Esik. On equations [21] B. Nagy. On the Membership Problem of Permutation for union-free regular languages. Inform. and Comput. Grammars – A Direct Proof of NP-Completeness. Int. 164/1 pp. 152–172, 2001. J. Found. Comput. Sci. 31/4 pp. 515–525, 2020. [2] J. Dassow and Gh. P˘ aun. Regulated rewriting in [22] B. Nagy and A. A. Akkeles. Trajectories and Traces on formal language theory. Springer-Verlag, Berlin, 1989. Non-traditional Regular Tessellations of the Plane. In: [3] V. Diekert and G. Rozenberg (eds.). The book of Combinatorial Image Analysis, 18th Int. Workshop, traces. World Scientific, River Edge, NJ, USA, 1995. IWCIA 2017, LNCS 10256, pp. 16–29, 2017. [4] K. Fogarasi and B. Nagy. A nondeterministic parser [23] B. Nagy and F. Otto. An Automata-Theoretical for Perm2 grammars. Abstract volume of the 10th Characterization of Context-Free Trace Languages. In: Joint Conference on Mathematics and Computer SOFSEM 2011: 37th Conference on Current Trends in Science (MaCS), Cluj-Napoca, Romania, 2014. Theory and Practice of Computer Science, LNCS [5] M. A. Harrison. Introduction to formal language 6543, pp. 406–417 2011. theory, Addison-Wesley, Reading, AM, USA, 1978. [24] B. Nagy and F. Otto. CD-systems of stateless [6] T. Herendi and B. Nagy. Parallel Approach of deterministic R(1)-automata governed by an external Algorithms. Typotex, Budapest, 2014. pushdown store. RAIRO Theor. Informatics Appl. 45 [7] R. Janicki, J. Kleijn, M. Koutny and L. Mikulski. pp. 413–448, (2011). Paradigms of Concurrency – Observations, [25] B. Nagy and S. Parchami. On deterministic sensing Behaviours, and Systems – a Petri Net View. Studies 0 0 5 → 3 Watson–Crick finite automata: a full hierarchy in Computational Intelligence 1020, Springer, 2022. in 2detLIN. Acta Informatica 58 pp. 153–175, 2021. [8] G. Madejski. Infinite hierarchy of permutation [26] S. Parchami and B. Nagy. Deterministic Sensing languages. 0 0 Fundam. Inform. 130/3 pp. 263–274, 2014. 5 → 3 Watson-Crick Automata Without Sensing [9] E. M¨ akinen. On permutative grammars generating Parameter. In: Unconventional Computation and context-free languages. BIT 25/4 pp. 604–610, 1985. Natural Computation, UCNC 2018, LNCS 10867, [10] A. Mateescu, G. Rozenberg and A. Salomaa. Shuffle Springer, pp. 173–187, 2018. on trajectories: syntactic constraints. Theoretical [27] M. Penttonen. One-sided and two-sided context in Computer Science 197 pp. 1–56, 1998. formal grammars. Inf. Control 25 pp. 371–392, 1974. [11] A. W. Mazurkiewicz. Trace Theory. In: Petri Nets: [28] R. Schott and J. C. Spehner. Two optimal parallel Central Models and Their Properties, Advances in algorithms on the commutation class of a word Petri Nets, Part II, LNCS, 255, pp. 279–324, 1986. Theoretical Computer Science 324 pp. 107–131, 2004. [12] B. Nagy. Union-free regular languages and [29] G. Rozenberg and A. Salomaa (eds.). Handbook of 1-cycle-free-path-automata. Publ. Math. Debrecen 68 formal languages. Springer-Verlag, Berlin, 1997. pp. 183–197, 2006. 38 Towards a Category-Theoretic Informatics Model of PSPP Linkages in Biomaterials Sylvert Prian Tahalea Miklós Krész sylvert@inf.u- szeged.hu miklos.kresz@innorenew.eu University of Szeged, Hungary InnoRenew CoE, UP IAM and UP FAMNIT, University of Universitas Pembangunan Nasional Veteran Yogyakarta, Primorska, Slovenia Indonesia University of Szeged, Hungary Abstract of formalism becomes a barrier when attempting to analyse the PSPP relationships at scale, particularly in data-driven or inter- This paper introduces a category-theoretic framework for mod- disciplinary contexts [25, 11]. eling the causal and structural relationships in the Processing- Recent advances in material informatics have highlighted the Structure-Property-Performance (PSPP) paradigm, with a case importance of formal representation for enabling computational study of bioluminescent bacterial interactions on engineered sur- reasoning, interoperability, and data integration [2, 24, 14, 7]. faces. The composition and interdependence of material behavior In parallel, category theory has emerged as a mathematical lan- in mathematical form are captured by defining the stages of PSPP guage for abstraction and composition, providing a rigorous way linkages into objects and their transformations as morphisms. to describe the structures and relationships between them [19]. This model is applied to a case study involving bacterial cul- Originally developed in mathematics, category theory has found tivation on three different engineered surfaces. The universal its way to be implemented in computer science, biology, data input configurations and multi-objective performance targets science, where it serves to represent the complex interactions are identified using limit and colimit, respectively. Additionally, [18, 4]. the functors and natural transformations applied compare the The category theory implementation in informatics offers a path- bacterial behaviors across material categories, illustrating the way to move beyond the classical representation of materials structural relationship between bacteria. While the current model knowledge, such as graphs and ontologies, to ensure the consis- assumes deterministic and discrete transitions, it opens pathways tency, scalability, and interoperability in computational models of for future development using enriched, fibred, and computational materials science. It has been recognised as a unifying language categorical tools. This work demonstrates the potential of cat- for science, offering tools such as objects, morphisms, limits, and egory theory as a unifying language for scientific modeling in colimits to formally capture relationships between complex sys- biomaterials research. tems [18, 4]. To address this gap, a categorical model is proposed Keywords in this paper to formalise the PSPP interconnections in a struc- tured and compositional way. Our contribution lies in defining category theory, PSPP, biomaterials, natural transformations, this formal representation, demonstrating how it can unify dis- functor parate descriptions into a coherent framework. The scope of this 1 work is focused on establishing the formal model rather than Introduction developing a full computational implementation. To illustrate The property-structure-processing-performance (PSPP) linkages its potential, a small example that highlights how categorical are a core paradigm in material science to conceptualise the in- structure captures PSPP linkages is provided. terdependence between material properties, processing methods, Section 2 The remainder of the paper is organised as follows. structural features, and resulting performance [12, 8, 1]. The PSPP section 3 explains the methodology, presents the case study linkages come from process-structure-property-performance re- section 4 example, concludes the contribution, limitations, and lationships, which build up like a chain in a bottom-up (simulation- future research. driven approach) or top-down (process design approach) [7]. Process refers to the methods and techniques used to transform 2 Methodology the raw materials into finished products with desired shapes This section explains the category, subcategories, functors, natu- and properties; structure refers to the arrangements of atoms, ral transformations, limits, and colimits of the PSPP linkages. molecules, or phases within the materials; properties are the mea- surable responses of a material to the stimuli and can be classified into various categories; while performance is the materials’ abil- 2.1 PSPP Linkage PSPP illustrates the chain of relationships that connect process- ity to function effectively in a specific application, considering its properties and conditions [7, 6, 10, 23]. Despite its importance, ing parameters to the resulting structure (microstructure), which PSPP linkages are described in an informal and domain-specific governs properties and thus impacts component performance—a manner, which hinders systematic integration across disciplines critical multiscale, multiphysics mapping essential for the model- and limits its computational implementation [15, 22]. This lack ing and design of materials [7]. Processing techniques, resultant structures, material qualities, and performance outcomes are all Permission to make digital or hard copies of all or part of this work for personal PSPP stages that are frequently examined separately or depicted or classroom use is granted without fee provided that copies are not made or informally using diagrams that lack formal semantics and math- distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this ematical rigor [7, 26, 3]. The PSPP framework helps to formalize work must be honored. For all other uses, contact the owner /author(s). the causal pathway from how a material is processed to its perfor- MATCOS-25, 9–10 October 2025, Koper, Slovenia mance outcome, e.g. bacteria-surface interaction for biolumines- © 2024 Copyright held by the owner/author(s). cence, which is later used as the study case. The bacteria-surface 39 MATCOS-25, 9–10 October 2025, Koper, Slovenia Tahalea & Krész interaction involves a complex and multilayered relationship be- luminescence output”. tween the processing method and the metabolic output. However, Define the morphisms in category PSPP as: C current approaches for modeling these interactions suffer from 𝑓 : 𝑃 𝑆 , , 𝑔 : 𝑆 𝑃 𝑟 , , and ℎ : 𝑃 𝑟 𝑃 𝑒 → → → several limitations, such as fragmented representation, lack of a unified framework, and lack of composability, which can be addressed with a category-theoretical model. The PSPP linkage Then the composite morphisms are: presented in Figure 1 shows how a specific processing method (e.g., oxygen plasma treatment) alters the surface structure, which 𝑔 𝑓 : 𝑃 𝑃 𝑟 and ℎ 𝑔 : 𝑆 𝑃 𝑒 ◦ → ◦ → in turn affects bacterial interaction properties (such as adhesion This composite morphism represents the entire PSPP causal chain or viability), ultimately leading to a measurable performance from processing to performance. For this case, it maps: outcome like bioluminescence intensity. “plasma treatment” “5 𝜇W/cm luminescence” −→ 2 via structure and property stages. The identity morphism at each stage (e.g., id : 𝑆 𝑆) satisfies: 𝑔 𝑔 id 𝑔, id 𝑟 𝑔 → ◦ = ◦ = 𝑆 𝑆 𝑃 confirming the category’s identity and associativity laws. 2.2.4 Diagram Representation. A typical morphism of the PSPP category is only represented by processing, structure, properties, Figure 1: PSPP linkages in bacteria-surface interaction. and performance. However, the processing can result in modified structure and properties, which are shown in Figure 2. 2.2 PSPP Category The category theoretic model of PSPP is defined as follows. 2.2.1 Objects. Let C be a category. The set of the objects in C is defined as: Ob 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 (C) = {} where: 𝑃 represents the manufacturing process or processing in short form, 𝑆 represents materials structure, 𝑃 𝑟 represents ma- terial properties, and 𝑃 𝑒 represents material performance. Thus, Ob encodes the fundamental entities of the PSPP framework (C) as objects within the categorical structure. Figure 2: PSPP diagram representation including modified 2.2.2 structure and properties. Morphisms. Define the set of morphisms (arrows) in the category as follows: C 𝐻 𝑜𝑚 𝑋 , 𝑌 ( C ) ( ) Where 𝑋 , 𝑌 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 and each morphism represents a re-∈ 2.3 Subcategories of PSPP Model lationship or transformation from one object to another. The The category model of PSPP linkage has several subcategories by morphisms in the category are defined as follows: nature (as presented in Figure 2), which are meaningful subsets C 𝑓 : 𝑃 𝑆 (Processing affect Structure) → of the objects and morphisms that still satisfy the definition of a category. 𝑔 : 𝑆 𝑃 𝑟 (Structure determine Properties) → ℎ : 𝑃 𝑟 𝑃 𝑒 (Properties dictate Performance) The main causal chain 𝑐𝑜𝑟 𝑒 → 2.3.1 Canonical Path Subcategory. C 𝑖 : 𝑃 𝑒 𝑃 (Performance feedback to Processing) → without branching. 𝑗 ′ • Objects. Ob(C 𝑐𝑜𝑟 𝑒 ) = 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 : 𝑃 → 𝑆 (Processing results Modified Structure) ′ • Morphisms. 𝑓 : 𝑃 → 𝑆, 𝑔 : 𝑆 → 𝑃 𝑟 , ℎ : 𝑃 𝑟 → 𝑃 𝑒 𝑘 : 𝑆 𝑃 𝑟 (Modified Structure determine Properties) → This paper will be focusing only on this canonical path or the 𝑙 : 𝑃 𝑟 𝑃 𝑟 (Properties transform to Modified Properties) → ′ main linkages. ′ → 𝑚 : 𝑃 𝑟 𝑃 𝑒 (Modified Properties dictate Performance) 2.3.2 Modified Path Subcategory. This modified subcategory 2.2.3 Composition. The composition of morphisms in the cate- C is an expansion of the canonical path that includes only 𝑚𝑜𝑑 gory follows the associativity and identity properties: the branching route. C ( ′ ′ ℎ ◦ 𝑔 ) ◦ 𝑓 = ℎ ◦ ( 𝑔 ◦ 𝑓 ) • Objects . Ob (C 𝑐𝑜𝑟 𝑒 ) = 𝑃 , 𝑆 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑟, 𝑃 𝑒 and for each object • Morphisms. ′ 𝑓 : 𝑃 → 𝑆 , 𝑗 : 𝑃 → 𝑆, 𝑔 : 𝑆 → 𝑃 𝑟 , 𝑋 ∈ Ob (C) , there exists an identity morphism ′ ′ ′ : 𝑘 𝑆 → 𝑃 𝑟 , 𝑙 : 𝑃 𝑟 → 𝑃 𝑟 , 𝑚 : 𝑃 𝑟 id → 𝑃 𝑒 𝑋 : 𝑋 → 𝑋 such that: id 𝑓 𝑌 ◦ = 𝑓 and 𝑔 ◦ id𝑋 = 𝑔 2.3.3 Feedback Loop Subcategory. This loop subcategory C 𝑙 𝑜𝑜 𝑝 focuses on feedback interactions within the model. In example for the bacterial surface interaction case, P = “oxy- gen plasma processing”, S = “hydrophilic glass surface”, Pr = . Ob 𝑐𝑜𝑟 𝑒 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 • Objects (C ) = 2 • Morphisms → “moderate bacterial adhesion, low EPS density”, Pe = “5 𝜇W/cm . 𝑓 , 𝑔, ℎ, 𝑖 : 𝑃 𝑒 𝑃 40 Category-Theoretic Modeling of PSPP Linkages MATCOS-25, 9–10 October 2025, Koper, Slovenia 2.3.4 Application Specific Subcategory. This subcategory can be Natural transformation of PSPP category model of our expanded based on the application focus field. The application Suppose there are 𝐹 , 𝐺 : 𝑐𝑜𝑛𝑐𝑟 𝑒𝑡 𝑒 two func-𝑏𝑎𝑐𝑡 use case: C → C on bacteria-surface interaction will be used as the case study in tors: this article, illustrated by following examples. • Pseudomonas flourescence 𝐹 : using • Biofilm subcategory C , with objects such as: • 𝐺: using 𝑏𝑖𝑜 𝑓 𝑖𝑙𝑚 Serratia marcescens 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 The natural transformation 𝑎𝑑ℎ𝑒𝑠𝑖𝑜𝑛 𝑏𝑖𝑜 𝑓 𝑖𝑙𝑚 𝜂 : 𝐹 ⇒ 𝐺, for each PSPP stage • Bioluminescence subcategory C , with objects such 𝑏𝑖𝑜𝑙𝑢𝑚 (object) transformation, comes from one species to another. as: 𝑃 , 𝑆 , 𝑃 𝑟 , 𝑃 𝑒 𝑙 𝑖𝑔ℎ𝑡 𝑙𝑢𝑚 2.5 Limit and Colimit 2.4 Functor and Natural Transformation 2.5.1 Cone. Let J be a category, C a category, and let 𝐷 : J → To define a functor and natural transformation of this model, first, C C be a functor (called a diagram in ). A cone over 𝐷 consists of: we need to define another category which can be processed with ∈ (C) (1) An object 𝑁 Ob , called the apex of the cone, this functor and natural transformation logic. This subsection was (2) A family of morphism in , 𝜙 𝑗 : 𝑁 𝐷 𝑗 , 𝑗 C { → ( )} ∈Ob( J ) prepared in advance to preserve the relation and transformation. → J such that for every morphism 𝑓 : 𝑗 𝑘 in , the follow- 2.4.1 Functor. ing commutative condition holds: 𝐷 A functor 𝐹 : ( C 𝑓 ) ◦ 𝜙𝑗 1 = → C 𝜙 2 between categories C 1𝑘 and In other words, a cone points into every object in the functor 2 C is defined by: (1) : for each object 𝑋 Ob 1 , there is ∈ (C ) Object mapping 𝐷 from the apex using the morphisms. object 𝐹 𝑋 Ob 2 Cone of PSPP model of our use case: ( ) ∈ (C ) Let there be three dif- (2) : for each morphism 𝑓 : 𝑋 𝑌 → ∈ Morphism mapping ferent processing routes for bioluminescent bacterial coating: Hom 𝐹 1: Processing 1, 𝐹2: Processing 2, and 𝐹3: Processing 3; which 𝑋 , 𝑌 C 1 ( ) , there is exist a morphism 𝐹 ( 𝑓 ) : 𝐹 ( 𝑋 ) → 𝐹 𝑌 Hom 𝐹 𝑋 , 𝐹 𝑌 ) ∈ C2 produce a full chain 𝑃𝑖 −→ ( ( ) ( )) − − → − − → ( 𝑓 𝑔 ℎ 𝑖 𝑖𝑖 𝑆𝑖 𝑃 𝑟𝑖 𝑃 𝑒𝑖 and the functor 𝐹 of such that: this output can be expressed as: 𝐹 : , where 𝐹 𝑗 𝑃 𝑒 . 𝑗 J → C ( ) = • Composition is preserved: 2.5.2 Limit. Let 𝐹 : J → C be a functor. A limit of 𝐹 is an object 𝐹 𝐿 Ob together with morphisms 𝜋 𝑗 : 𝐿 ∈ (C) ( 𝑓 ◦ 𝑔 ) = 𝐹 ( 𝑓 ) ◦ 𝐹 ( 𝑔 ) , ∀ 𝑓 , 𝑔 ∈ Mor (C 1 ) → 𝐹 ( 𝑗 ), ∀𝑗 ∈ Ob ′ such that for every morphism (J ) , 𝑓 : 𝑗 → 𝑗 in J , • Identities are preserved: 𝐹 𝑓 𝜋 𝑗 𝜋𝑗 ′ ( ) ◦ = 𝐹 id𝑋 id , 𝐹 ( ) = ( 𝑋 ∀𝑋 ∈ Mor )(C1) Moreover, 𝐿, 𝜋 : for any other cone 𝑗 ( ) is universal {𝜓 : 𝑗 𝑁 → Functor of PSPP category model example in our use case: 𝐹 ( 𝑗 )} , there exists a unique morphism 𝑢 : 𝑁 𝑗 ∈ Ob → 𝐿 such ( J ) Considering two distinct PSPP categories: that • C : a category of PSPP stages for 𝑔𝑙 𝑎𝑠𝑠 Pseudomonas flourescens 𝜓𝑗 = 𝜋𝑗 ◦ 𝑢, ∀𝑗 ∈ Ob(J ). treated on surface. glass Limit of PSPP Model of our use case: The output of the • C : a category of PSPP stages for 𝑤𝑜𝑜𝑑Pseudomonas flourescens PSPP linkage lies at the performance stage; the subdiagram from on functionalized . wood processing to performance can be used to find the optimal so- Then we can define Functor 𝐹 : : lution through the PSPP model. Assume that there are three 𝑔𝑙 𝑎𝑠𝑠 C → C 𝑤𝑜𝑜𝑑 • Objects different processing routes: (1) 𝐹1 : 𝑃1 → 𝑆1 → 𝑃 𝑟1 → 𝑃 𝑒1, (2) – → → 𝑃 𝐹 2 : 𝑃2 𝑆2 𝑃 𝑟 𝑃 enzym-2 → 𝑃 𝑒2, and (3) 𝐹 1 : 𝑃3 → 𝑆3 → 𝑃 𝑟3 → 𝑃 𝑒3. 𝑔𝑙 𝑎𝑠𝑠 → : oxygen plasma cleaning 𝑤𝑜𝑜𝑑 → The limit object 𝐿 is universal, thus there exist projections to based-pretreatement – each case: 𝑆 𝑔𝑙 𝑎𝑠𝑠 → 𝑆 : smooth hydropphilic silica porous 𝑤𝑜𝑜𝑑 → lignovellulosic surface 𝜋𝑖 : 𝐿 𝑃 𝑒𝑖 for 𝑖 1, 2, 3 → = – 𝑃 𝑟 𝑃 𝑟 : low biofilm density 𝑔𝑙 𝑎𝑠𝑠 → 𝑤𝑜𝑜𝑑 → moderate Let 2.5.3 Colimit. 𝐹 : J → C be a functor. A colimit of 𝐹 is biofilm density an object 𝐶 Ob together with morphisms 𝜄 : 𝑗 ∈ (C) ( ) → 𝐹 𝑗 – 𝑃 𝑒 𝑃 𝑒 𝑔𝑙 𝑎𝑠𝑠 → : weak bioluminescence 𝑤𝑜𝑜𝑑 → stable ′ 𝐶, ∀ 𝑗 ∈ Ob (J ) , such that for every morphism 𝑓 : 𝑗 → 𝑗 in J , 𝜄 ′ 𝐹 𝑓 𝜄 . Moreover, 𝐶, 𝜄 is : for any other cocone 𝑗 𝑗 𝑗 • Morphisms {𝜓𝑗 : 𝐹 ( 𝑗 ) → 𝑁 } , there exists a unique morphism 𝑣 : 𝑗 ∈ Ob ( J ) – bioluminescence overtime ◦ ( ) = ( ) universal 𝑓 𝑓 : plasma alters hydrophilicity enzy-→ → 𝑔𝑙 𝑎𝑠𝑠 𝑤𝑜𝑜𝑑 𝐶 → = ◦ ∀ ∈ (J ) 𝑁 such that 𝜓 𝑗 𝑣 𝜄 𝑗 , 𝑗 Ob . matic treatement alters lignin exposure Colimit of PSPP Model of our use case: In the study of the – 𝑔 𝑔 rough-𝑔𝑙 𝑎𝑠𝑠 → : smooth surface limit adhesion 𝑤𝑜𝑜𝑑 → PSPP linkage, suppose that there are three different properties, ness enhances microbial anchoring as follows. (1) 𝐹 : 1 𝑃 𝑟𝐴 → 𝑃 𝑒 , (2) , (3) 𝐴 𝐹 : 2 𝑃 𝑟 𝐵 𝑃 𝑒𝐵 𝐹 : → 3 – ℎ𝑔𝑙 𝑎𝑠𝑠 → ℎ : low metabolic activity 𝑤𝑜𝑜𝑑 → higher growth 𝑃 𝑟 ; one of the objectives is to merge or combine all → 𝐶 𝑃 𝑒 𝐶 and igh emission the performances to produce a unified performance outcome – 𝑖𝑔𝑙 𝑎𝑠𝑠 → 𝑖 : bioluminescence feedback informas 𝑤𝑜𝑜𝑑 𝜄 𝑖 : 𝑃 𝑒𝑖 → 𝐶 and any projection results factors uniquely through wood resin optimization 𝐶. Having established the functor, a later section presents the use case of the PSPP model. 3 Case Study: Optimising Surface Design for 2.4.2 Bioluminescent Bacteria Natural Transformation. Let 𝐹 , 𝐺 : C 1 → C 2 be functors. A natural transformation 𝜂 : 𝐹 𝐺 is a family of morphisms 𝜂 : The case study of this model is to determine the best surface con-⇒ 𝑋 𝐹 𝑋 𝐺 𝑋 , 𝑋 Ob 1 , such that for every morphism figuration to support bioluminescence output from ( ) → ( ) ∀ ∈ (C ) Pseudomonas 𝑓 : 𝑋 𝑌 in 1, 𝜂𝑌 → C ◦ 𝐹 ( 𝑓 ) = 𝐺 (𝑓 ) ◦ 𝜂 . 𝑋 flourescens. There are three different surface preparation routes 41 MATCOS-25, 9–10 October 2025, Koper, Slovenia Tahalea & Krész [16, 13, 17, 9, 20, 5, 21] using the PSPP framework and interpret- 3.4 Colimit Construction ing the model categorically, such as glass, wood, and concrete. Conversely, a colimit object 𝐶 which integrates all outcomes into a general-purpose performance target should be defined as presented in Figure 5. 𝐶 serves as a composite performance 3.1 Experimental Pipelines Each processing route is a distinct PSSP chain. (1) Oxygen plasma on glass: 𝑃 : Oxygen plasma cleaning; : Enhanced surface hy-1 𝑆 1 drophilicity and cleanliness; 𝑃 𝑟 1: Reduced adhesion of bacteria; 𝑃 𝑒 : Produce a low bioluminescence signal caused by the low 1 density of biofilm. (2) Enzyme treatment on wood: 𝑃2: Enzymatic pretreatment of the surface; 𝑆 2: Surface may degrade and reduce the cohesion; 𝑃 𝑟 2: Low biofilm viability; 𝑃 𝑒2: Low signal output due to diminished bacterial presence. (3) Sol-gel coated concrete: 𝐶 Figure 5: A cocone with apex absorbing all performances 𝑃 3: Sol-gel encapsulation; 𝑆3: Potentially surrounded by silica output gel, contains methanol; 𝑃 𝑟 3: Severely limits bacteria’s metabolic activity; 𝑃 𝑒 : Light output reduced drastically. The PSPP linkage 3 specification, capturing multi-objective behaviors (e.g., balancing on this study case is presented in Figure 1. intensity and longevity) with the following categorical properties. • Morphisms 𝜄𝑖 : 𝑃 𝑒𝑖 → 𝐶 3.2 Categorical diagram • 𝐶 is the colimit of every projection from 𝑃 𝑒 factors uniquely 𝑖 Given that each route forms a PSPP morphism chain: through 𝐶 𝑓 𝑔 ℎ 𝑖 𝑖𝑖 𝑃𝑖 𝑆 , which is presented in Figure 3. −→ 𝑖 − − → 𝑃 𝑟 𝑖 − − → 𝑃 𝑒 𝑖 3.5 Natural transformation Suppose there is another experiment with the same three surfaces but different bacteria, i.e. . The functors and Serratia marcescens the natural transformation are presented in Figure 6 and can be defined as follows. • Functors: 𝐹: PSPP route using P. flourescens and 𝐺: PSPP route using S. marcescens • Natural transformation: 𝜂 : 𝐹 ⇒ 𝐺, where 𝜂𝑃: translate the process (e.g plasma treatment) parameters, 𝜂𝑆: map structure adaptation, and 𝜂𝑃 𝑟 , 𝜂𝑃 𝑒: transform properties and performances across bacterial strains. Figure 3: The routes of PSPP morphism chain 3.3 Limit Construction A universal configuration 𝐿 (e.g., a hybrid substrate combining features of all three processes) is defined as a cone in Figure 4. Universal configuration 𝐿 represents an ideal design that projects Figure 6: Natural transformation from P. flourescence to S. marcescens 3.6 Discussion The categorical analysis of PSPP linkages in bioluminescent bac- terial systems provides a structured lens to evaluate and compare multiple experimental design routes. In this case study, three Figure 4: 𝐿 is the universal input that relates to all observed distinct processing strategies—oxygen plasma treatment on glass, performances enzyme pretreatment on wood, and sol-gel silica encapsulation on concrete—were formalised as morphism chains within the to each observed performance, supporting general optimization. C category . Each route defines a unique instance of a func- PSPP The categorical properties are as follows. → tor mapping the canonical PSPP stages (Processing Structure • 𝐿 → 𝑃 𝑒 form morphims → Property → Performance). The resulting categorical diagram 𝑖 • Commutativity must hold: any internal morphism be- allowed us to analyse their behaviours both individually and tween 𝑃 𝑒 must be maintained if they exist relationally. 𝑖 • 𝐿 is the limit and universal: every cone factors uniquely The introduction of a limit object 𝐿, structured as a cone over through 𝐿 the performance outcomes 𝑃 𝑒1, 𝑃 𝑒2, and 𝑃 𝑒3, offered a formal 42 Category-Theoretic Modeling of PSPP Linkages MATCOS-25, 9–10 October 2025, Koper, Slovenia approach to identifying an optimal or universal input configura- [3] Igor Baskin and Yair Ein-Eli. 2022. Electrochemoinformatics as an emerging scientific field for designing materials and electrochemical energy storage tion capable of reproducing all observed performance behaviors. and conversion devices—an application in battery science and technology. Practically, 𝐿 can be interpreted as a generalized hybrid surface Advanced Energy Materials , 12, 48, 2202380. doi: 10.1002/aenm.202202380. or processing strategy that maintains compatibility with the [4] Brendan Fong and David I Spivak. 2018. Seven sketches in compositionality: an invitation to applied category theory. . arXiv preprint arXiv:1803.05316 range of observed bioluminescence behaviors. Conversely, the doi: 10.48550/arXiv.1803.05316. dual construction of a object 𝐶, defined via morphisms [5] Wissam Ghach, M. Etienne, V. Urbanová, Frédéric P. A. Jorand, and A. Wal-colimit 𝜄 𝑖 : 𝑃 𝑒𝑖 carius. 2014. Sol–gel based ‘artificial’ biofilm from pseudomonas fluorescens → 𝐶 , enabled the synthesis of a composite performance using bovine heart cytochrome c as electron mediator. Electrochemistry profile, which may serve as a design target in multi-objective Communications, 38, 71–74. doi: 10.1016/J.ELECOM.2013.11.001. optimization tasks. [6] D. Gu, Xinyu Shi, R. Poprawe, D. Bourell, R. Setchi, and Jihong Zhu. 2021. Material-structure-performance integrated laser-metal additive manufac- This dual analysis captures generalisation (limit) and integration turing. , 372. doi: 10.1126/science.abg1487. Science (colimit), offering a flexible modelling approach. The model sup- [7] Seyed Mahdi Hashemi, Soroush Parvizi, Haniyeh Baghbanijavid, Alvin TL ports the comparative approach through natural transformation Tan, Mohammadreza Nematollahi, Ali Ramazani, Nicholas X Fang, and Mohammad Elahinia. 2022. Computational modelling of process–structure– between bacteria through functors. For example, transitions be- property–performance relationships in metal additive manufacturing: a tween and can be review. International Materials Reviews Pseudomonas fluorescens Serratia marcescens, 67, 1, 1–46. doi: 10.1080/09506608.2 formalised as structure-preserving morphism families, highlight- 020.1868889. [8] Surya R Kalidindi and Marc De Graef. 2015. Materials data science: current ing how microbial adaptation maps across processing contexts. Annual Review of Materials Research status and future outlook. , 45, 1, 171– This indicates category theory’s ability to enable reasoning and 193. doi: 10.1146/annurev- matsci- 070214- 020844. [9] A. Kotlobay et al. 2018. Genetically encodable bioluminescent system from potential computational automation in material informatics. fungi. Proceedings of the National Academy of Sciences of the United States of America, 115, 12728–12732. doi: 10.1073/pnas.1803615115. 4 Conclusion [10] Wenqi Li and Jian Shi. 2023. Lignin-derived carbon material for electro- chemical energy storage applications: insight into the process-structure- With a brief case study of bioluminescent bacterial interactions Frontiers in Bioengineering and Biotech- properties-performance correlations. on artificial surfaces, this paper offers a category-theoretic frame- , 11. doi: 10.3389/f bioe.2023.1121027. nology [11] Zhanzhao Li, Te Pei, Weichao Ying, Wil V Srubar III, Rui Zhang, Jinyoung work for modelling PSPP linkage in biomaterials. In order to allow Yoon, Hailong Ye, Ismaila Dabo, and Aleksandra Radlińska. 2024. Can do- compositional thinking, a formal categorical structure is main knowledge benefit machine learning for concrete property prediction? PSPP C designed to represent PSPP stages as objects and causal relation- , 107, 3, 1582–1602. doi: 10.1111/jac Journal of the American Ceramic Society e.19549. ships as morphisms. The model captures both variability and [12] Gregory B Olson. 1997. Computational design of hierarchically structured unification across various experimental settings by using func- materials. , 277, 5330, 1237–1242. doi: 10.1126/science.277.5330.1237. Science [13] Rahman Rahmanpour and T. Bugg. 2015. Characterisation of dyp-type per- tors, natural transformations, and universal constructions like oxidases from pseudomonas fluorescens pf-5: oxidation of mn(ii) and poly- limits and colimits. Archives of biochemistry and biophysics meric lignin by dyp1b. , 574, 93–8. A case study of across three surface doi: 10.1016/j.abb.2014.12.022. Pseudomonas fluorescens [14] Balashanmuga Priyan Rajamohan et al. 2025. Materials data science ontol- treatments shows the effectiveness of this approach in deter- ogy (mds-onto): unifying domain knowledge in materials and applied data mining optimal input configurations (by limits) and composite science. Scientific Data, 12, 1, 628. doi: 10.1038/s41597- 025- 04938- 5. performance targets (via colimits). The application of natural [15] Krishna Rajan. 2005. Materials informatics. Materials Today, 8, 10, 38–45. doi: 10.1016/S1369- 7021(05)71123- 8. transformations furthers biological comparison across strains. [16] Z. Remeš, O. Babčenko, Vítězslav Jarý, and K. Beranová. 2024. Enhanced pho- The main contributions of this work are: (1) a formal mathemati- toluminescence of plasma-treated recycled glass particles. Nanomaterials, 14. doi: 10.3390/nano14131091. cal structure for PSPP links based on category theory and (2) the [17] Francis W. M. R. Schwarze et al. 2024. Taming the production of biolumi- integration of experimental design with universal structures like Advanced nescent wood using the white rot fungus desarmillaria tabescens. cones and cocones. However, there are some limits. The current , 11. doi: 10.1002/advs.202403215. Science [18] David I Spivak. 2014. . MIT press. Category theory for the sciences model assumes discrete and predictable transitions between PSPP [19] David I Spivak, Tristan Giesa, Elizabeth Wood, and Markus J Buehler. 2011. phases, does not account for uncertainty, dynamics, or probabilis- Category theoretic analysis of hierarchical protein materials and social tic behavior, and lacks an implemented computational instance networks. , 6, 9, e23911. doi: 10.1371/journal.pone.0023911. PloS one [20] Aisha J Syed and James C. Anderson. 2021. Applications of bioluminescence functor to real data. in biotechnology and beyond. . doi: 10.1039/d0cs01 Chemical Society reviews Future work will focus on enriching the categorical model with 492c. [21] J. Trögl, G. Kuncová, and P. Kurán. 2010. Bioluminescence of pseudomonas quantitative structure (e.g., cost, confidence, time) via enriched fluorescens hk44 in the course of encapsulation into silica gel. effect of categories; defining pullbacks and pushouts for constraints. Folia Microbiologica methanol. , 55, 569–575. doi: 10.1007/s12223- 010- 0091- 9. Acknowledgements [22] Kewei Wang et al. 2024. Deep learning based inverse modeling for materials 2024 international design: from microstructure and property to processing. In The research was supported by the BioLOG project: the second . IEEE, 236–241. conference on machine learning and applications (ICMLA) doi: 10.1109/ICMLA61862.2024.00038. author is grateful for the support of the National Centre of Science [23] Yayun Wang, S. Naleway, and Bin Wang. 2020. Biological and bioinspired (NCN) through grant DEC-2020/39/I/HS4/03533, the Slovenian materials: structure leading to functional and mechanical performance. Research and Innovation Agency (ARIS) through grant N1-0223, , 5, 745–757. doi: 10.1016/j.bioactmat.2020.06.003. Bioactive Materials [24] Tian Xie and Jeffrey C Grossman. 2018. Crystal graph convolutional neural and the Austrian Science Fund (FWF) through grant I 5443-N. This networks for an accurate and interpretable prediction of material properties. work is supported by the ARIS research program P1-0404 and by , 120, 14, 145301. doi: 10.1103/PhysRevLett.120.145301. Physical review letters [25] Han Zhang, Runsheng Li, Junjiang Liu, Kaiyun Wang, Qian Weijian, Lei Shi, the research program CogniCom (0013103) at the University of Liming Lei, Weifeng He, and Shengchuan Wu. 2024. State-of-art review on Primorska. the process-structure-properties-performance linkage in wire arc additive manufacturing. , 19, 1, e2390495. doi: 10.10 Virtual and Physical Prototyping References 80/17452759.2024.2390495. [26] Olga Zinovieva, Varvara Romanova, Ekaterina Dymnich, Aleksandr Zi- [1] Ankit Agrawal and Alok Choudhary. 2016. Perspective: materials informat- noviev, and Ruslan Balokhonov. 2023. A review of computational approaches ics and big data: realization of the “fourth paradigm” of science in materials to the microstructure-informed mechanical modelling of metals produced science. , 4, 5. doi: 10.1063/1.4946894. Apl Materials Materials by powder bed fusion additive manufacturing. , 16, 19, 6459. doi: [2] Toshihiro Ashino. 2010. Materials ontology: an infrastructure for exchanging 10.3390/ma16196459. materials information and knowledge. , 9, 54–61. doi: Data Science Journal 10.2481/dsj.008- 041. 43 Cost-Sensitive Overview of Model Ensembling for Machine-Generated Text Detection Abstract both during training and inference, which can limit the practical In this paper, we compare different ways of combining encoder- adoption of such methods in large-scale, real-time applications. based neural language models for machine-generated text detec- To address these concerns, this paper provides a cost-centric tion from a cost-centric perspective. We tested five ensembling overview of different strategies adapted for machine-generated approaches: soft voting with separately fine-tuned models, train- text detection. For our experiments, we relied on a recent publicly ing ensembles on disjoint training data subsets (for reducing available dataset: Task 1, Subtask A of the Workshop on Detecting training costs), snapshot ensembles, models with multiple classi- AI-Generated Content at COLING 2025 (Wang et al., 2025), where fication heads, and merging models by averaging their weights. the primary goal is to classify English text as human-written or We evaluated each method based on how accurate they are (using AI-generated. We compare a range of approaches, including soft- macro F1 scores) and how much computing costs they require voting ensembles, ensembles on disjoint training data subsets, during training and prediction. Our findings show that while snapshot ensembles, multiple classification heads model, and typical ensemble methods can boost accuracy, they come with weight space model merging. By evaluating each method not only high resource usage. In comparison, model merging achieves on predictive accuracy, but also on their training and inference the highest accuracy (macro F1 of 0.832) without increasing the costs, we aim to identify solutions that offer a practical balance inference time. Although model merging requires higher training between effectiveness and efficiency. costs, this is often less of a concern where inference time costs Our results show that, while traditional ensembling improves outweigh the costs of model construction. performance, it also results in significant computational over- head both during training and inference time. In contrast, model Keywords merging, where multiple models’ weights are combined, achieves ensemble methods, ai text detection, encoder models higher accuracy without increasing inference costs compared to a single model. Although training becomes more expensive, this 1 is a one-time expense, making model merging an efficient and Introduction In recent years, improvements in generative models have made for.anonimity. practical solution. Our source code is available at http://retracted. machine-generated text much more natural and realistic. Large volumes of AI-produced content are now being created and have even found their way into academic and scientific publica- which can now be accessed by users with minimal technical tions (Liang et al., 2024b). For example, previous studies indicate 2 Related Work that as much as 6.5% to 16.9% of peer review submissions to AI Recent advancements in detecting machine-generated text have conferences may be heavily influenced or rewritten by large lan-highlighted the efficacy of ensemble learning methods, partic-guage models, highlighting a growing dependence on LLMs in ularly those that employ soft voting strategies (Gu and Meng, scholarly environments (Liang et al., 2024a). This trend is further 2024; Kiss and Berend, 2025). In the SemEval-2024 Task 8 (Wang reinforced by the widespread availability of powerful models, et al., 2024), Gu and Meng (2024) introduced a class-balanced soft voting system that fine-tuned transformer-based models (Gu and expertise. Meng, 2024). Their approach effectively addressed data imbal-Because of this, there is a growing need for automated tools ance and achieved state-of-the-art performance in multi-class that can distinguish between human-written and machine-generated classification tasks involving various text generators. text. This need has been underscored by several recent shared Due to the high computational cost of standard ensembles, tasks and competitions focused on this problem (Sarvazyan et al., which require training multiple independent models, several ap-2023; Wang et al., 2024; Chamezopoulos et al., 2024; Wang et al., proaches have been proposed to reduce resource usage. One such 2025). method is the snapshot ensemble technique (Huang et al., 2017), Even with notable progress in the field, reliably separating which captures multiple model “snapshots” at different points machine-generated and human-written content remains difficult. during a single training run. Although this approach decreases Most state-of-the-art approaches for detecting AI-generated text training costs by avoiding the need to train separate models, it rely on deep neural architectures, particularly transformer-based usually does not reach standard ensemble performance. models fine-tuned on large annotated datasets (Sarvazyan et al., In the domain of multi-label and large-scale classification, 2024; Marchitan et al., 2024). Recent research results have shown Liang et al. (2025) proposed the Multi-Head Encoding (MHE) that the combination of multiple models can further improve the approach to tackle extreme label classification challenges(Liang accuracy and robustness of machine-generated text detection et al., 2025). Their innovation involves substituting a conven-(Gu and Meng, 2024; Kiss and Berend, 2025). However, these tional single classifier with multiple specialized classification gains come at the expense of increased computational demands, heads, with each head responsible for a distinct subset of the work must be honored. For all other uses, contact the owner/author(s). Permission to make digital or hard copies of all or part of this work for personal overall label space. Their experiments across several extreme or classroom use is granted without fee provided that copies are not made or classification benchmarks underscore the potential of multiple distributed for profit or commercial advantage and that copies bear this notice and classification head architectures in improving both efficiency the full citation on the first page. Copyrights for third-party components of this and generalization, which aligns with our motivation for using Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia models with multiple classification heads in the context of AI- © 2024 Copyright held by the owner/author(s). generated text detection. 44 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Trovato et al. 3 Methodology 3.4 Snapshot Ensemble 3.1 Base models For the snapshot ensemble approach (Huang et al., 2017), we gen- provides more than 610,000 English texts for training and 261,000 Dataset. erated several “snapshots” from a single model taken at different The shared task dataset (Wang et al., 2025) we used points throughout its fine-tuning. Unlike traditional ensemble methods, which require training multiple independent models for validation. Each entry is labeled either machine-generated by from scratch, snapshot ensembling takes advantage of a cyclical one of 40 different LLMs or human-written, yielding a 41-class learning rate schedule to encourage the model to converge to classification task. diverse regions of the parameter space within a single training written text as a single, heterogeneous category. Each DeBERTa-three separate DeBERTa-base models (He et al., 2021) using a 41-Single Model. run. This makes the method much more resource efficient, as it For our ensemble experiments, we fine-tuned avoids the repeated cost of training from random initializations. In our experiments, we used a cyclical learning rate schedule, class setup. By adopting this multi-class approach, the models are as described in the original method. This approach periodically better at capturing the diversity and differences among texts gen- increases and decreases the learning rate during training. At the erated by different models, rather than treating all non-human end of each cycle, we saved the model as a snapshot (see Figure 1). We decided to save six snapshots to achieve sufficient diversity. base model was trained independently using the AdamW opti- mizer (learning rate 2e-5, weight decay 0.01), with a batch size of 16, a 10% warmup schedule, and early stopping based on vali- dation performance. In our work, we intentionally chose not to rely on generative large language models (LLMs) as the core detection mechanism. This decision was motivated by practical considerations of cost efficiency and scalability. LLM-based solutions often have high computational costs and can be difficult to deploy at scale. As a result, our approach prioritizes methods that are both accurate and cost-effective. 3.2 Figure 1: Learning rate schedule for snapshot ensemble Vanilla Ensemble works well for machine-generated text detection (Abburi et al., to the final prediction, are widely used to improve both accuracy snapshots (based on validation performance) and combined their predictions via soft voting. This approach allowed us to obtain and robustness in machine learning (Ganaie et al., 2022). It also multiple strong models within a single fine-tuning process. How- Traditional ensemble methods, also known as output-space en-sembling, where several independently trained models contribute To construct the ensemble, we selected the three most diverse where each model was fine-tuned separately, and their predicted In our experiments, we used a standard soft voting approach, sembles. Although the improvement in predictive performance over a single model was modest, the snapshot ensemble presents class probabilities were averaged to determine the final output. a practical alternative when training resources are limited. Up to 2023; Gu and Meng, 2024; Kiss and Berend, 2025). the overall inference cost remains similar to that of standard en- ever, since inference requires running each selected snapshot, This typically led to better classification results than any single this point, the examined solutions mainly focused on reducing model. training costs, without lowering inference demands. From now However, this improvement comes with significant resource on, we explore methods that aim to improve inference efficiency costs as all models must be trained and evaluated individually. as well. For our study, we fine-tuned three models and combined their predictions using soft voting, then compared the ensemble’s 3.5 Model Merging macro F1 score with the average score of the single models. Model merging, also known as weight space ensembling or model soups (Wortsman et al., 2022), is a technique explored in machine 3.3 Disjoint Training Data (DTD) Ensembles learning as an alternative to traditional output space ensembling. As we wanted to keep the total fine-tuning cost of the ensemble Although assembling multiple independently trained models by close to that of a single model, while still benefiting from the averaging their predictions can improve performance, it comes at advantages of ensembling, we employed an ensemble method, the cost of increased inference computation, as the output of sev- where we trained the model on disjoint training data subsets. eral models must be calculated. Model merging addresses this by Knowing the large size of our available training data (approx. averaging the weights of several models to form a single, merged 611,000 examples), we divided the dataset into three disjoint model. This merged model can match or exceed the performance subsets of equal size. Each subset was independently used to of output space ensembles in many cases, but crucially, it incurs fine-tune a separate model, resulting in three models trained no additional inference costs compared to a single model. on disjoint portions of the dataset. The smaller training subsets The effectiveness of weight space merging is supported by two shorten the individual training durations, making the approach fundamental concepts: mode connectivity (Frankle et al., 2020) more efficient overall. and sparsity. Mode connectivity suggests that independently After training, the three models were combined into an ensem- trained neural networks, especially those derived from the same ble using soft voting. This setup uses the diversity of individually base model, can reside in regions of the loss landscape connected trained models while offering a cost-effective and time-efficient by paths of relatively constant performance. In this work, we training alternative to standard ensemble strategies. applied a model merging approach using three distinct models. 45 Cost-Sensitive Overview of Model Ensembling for Machine-Generated Text Detection Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia After merging, we experimented with two configurations. remained high due to querying multiple models simultaneously. First, we attached the original three classification heads from The Snapshot Ensemble offered moderate computational effi- the source models to the merged encoder, resulting in three sep- ciency, yet resulted in a small improvement of macro F1 over arate models sharing the same backbone but differing in their the single model, while still retaining high inference costs due classification heads. Second, we also fine-tuned new classification to combining predictions from multiple snapshots. MCH models heads on top of the merged encoder, which model variant we demonstrated varying performance depending on the number of refer to as Model Merging (FT). This setup kept the efficiency of heads. The 3-head variant was matching single-model costs but a single-model inference pipeline while allowing the new head did not outperform it. The 5-head variant improved F1 to 0.817 to adapt to the merged weights. while maintaining low inference costs and moderate training ex- pense. Model merging is the most promising approach. Directly 3.6 Models with Multiple Classification Heads merging pre-trained model weights showed the best overall per- In this architecture, multiple classification heads are attached to a formance, maintaining low inference cost equivalent to a single shared backbone model (Chang et al., 2023; Lee et al., 2015). Each model. Further fine-tuning after merging increased training costs, head is independently fine-tuned, allowing them to specialize but unlike ensembles, the merged model requires only a single and capture different aspects of the data distribution. During model at inference. This greatly reduces inference time and re- inference, the outputs of these heads are aggregated, typically by source usage. Considering both computational efficiency and voting to produce the final prediction. This approach provides classification performance, the model merging strategy without some of the benefits of ensembling, while maintaining a compact additional fine-tuning provided the best balance overall. model structure and reducing deployment complexity. In our experiments, we implemented variants with 3 and 5 classification 5 Conclusion heads (referred to as MCH models). Each head independently pro- We explored multiple ensemble strategies for detecting machine- duced a classification output, and the final label was determined generated text, highlighting a cost-sensitive approach that bal- by soft voting over them. The main advantage of the multiple ances computational efficiency with predictive performance. Our classification headed (MCH) architecture is its low inference cost, evaluation demonstrated that although ensemble methods consis- equivalent to a single model since all heads share the same en- tently improve predictive accuracy, they typically require signifi- coder. The only additional overhead comes from running the cantly higher training or inference resources, making them less multiple classification heads, which is negligible compared to practical for real-world scenarios where resources are limited. In vanilla multi-model ensembles. particular, model merging was the most effective option among all the evaluated methods. Not only did it achieve the highest 4 Results macro F1 score (0.832), but it did so without inflating inference We compared several ensembling approaches, focusing on both costs, maintaining an efficiency comparable to running just a their Macro-F1 scores and their training and inference costs. The single model. This approach effectively captures the strengths vanilla ensemble and the fine-tuned model merging approach are of multiple models while avoiding the computational overhead averaged over three independent runs, the rest are based on a commonly associated with traditional ensembling methods. single evaluation. Overall, our results show that the use of ensembles does not always have to be very costly. By carefully choosing methods Experiment Costs such as model merging, it is possible to get most of the perfor- mance benefits of ensembling without using too many resources Macro F1 Training Inference for predictions, especially during inference time. This can make 3 Classifier Heads 0.805 Low Low these methods a good fit for building fast and scalable systems Single model to detect AI-generated text. 0.806 Low Low Snapshot ensemble 0.810 Low High DTD Ensemble 0.814 Low High References 5 Classifier Heads 0.817 Low Low Harika Abburi, Kalyani Roy, Michael Suesserman, Nirmala Pu- Vanilla Ensemble 0.826 High High dota, Balaji Veeramani, Edward Bowen, and Sanmitra Bhat- Model Merging for AI-generated Text Detection. arXiv:2311.03084 [cs.CL] 0.832 High Low https://arxiv.org/abs/2311.03084 Model Merging (FT) tacharya. 2023. A Simple yet Efficient Ensemble Approach 0.827 Highest Low Table 1: Summary of results on test set with costs. Savvas Chamezopoulos, Drahomira Herrmannova, Anita De Waard, Drahomira Herrmannova, Domenic Rosati, and Our experimental results are summarized in Table 1, which Yury Kashnitsky. 2024. Overview of the DagPap24 Shared compares the predictive performance (macro F1 score) and com- Task on Detecting Automatically Generated Scientific Paper. putational costs (training and inference) of each ensemble method. In Proceedings of the Fourth Workshop on Scholarly Document The Vanilla Ensemble achieved an improved macro F1 score of Processing (SDP 2024), Tirthankar Ghosal, Amanpreet Singh, 0.826, which was a substantial increase over the single model. Anita Waard, Philipp Mayr, Aakanksha Naik, Orion Weller, However, this performance gain incurred significantly higher Yoonjoo Lee, Shannon Shen, and Yanxia Qin (Eds.). Association costs for both training and inference. for Computational Linguistics, Bangkok, Thailand, 7–11. The DTD ensemble approach resulted in a slightly lower macro https://aclanthology.org/2024.sdp-1.2 F1 compared to the vanilla ensemble. However, it significantly Haw-Shiuan Chang, Ruei-Yao Sun, Kathryn Ricci, and Andrew reduced training costs, since each individual model was trained McCallum. 2023. Multi-CLS BERT: An Efficient Alternative on only one-third of the training data, although inference costs to Traditional Ensembling. In Proceedings of the 61st Annual 46 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Trovato et al. Meeting of the Association for Computational Linguistics (Vol- Weixin Liang, Yaohui Zhang, Zhengxuan Wu, Haley Lepp, Wen- ume 1: Long Papers), Anna Rogers, Jordan Boyd-Graber, and long Ji, Xuandong Zhao, Hancheng Cao, Sheng Liu, Siyu He, Naoaki Okazaki (Eds.). Association for Computational Linguis- Zhi Huang, Diyi Yang, Christopher Potts, Christopher D Man- tics, Toronto, Canada, 821–854. https://doi.org/10.18653/v1/ ning, and James Y. Zou. 2024b. Mapping the Increasing Use 2023.acl-long.48 of LLMs in Scientific Papers. In First Conference on Language Jonathan Frankle, Gintare Karolina Dziugaite, Daniel M. Roy, and Modeling. https://openreview.net/forum?id=YX7QnhxESU Michael Carbin. 2020. Linear mode connectivity and the lot- Teodor-george Marchitan, Claudiu Creanga, and Liviu P. Dinu. tery ticket hypothesis. In Proceedings of the 37th International 2024. Team Unibuc - NLP at SemEval-2024 Task 8: Trans- Conference on Machine Learning (ICML’20). JMLR.org, Article former and Hybrid Deep Learning Based Models for Machine- 305, 11 pages. Generated Text Detection. In Proceedings of the 18th Inter- M.A. Ganaie, Minghui Hu, A.K. Malik, M. Tanveer, and P.N. Sug- national Workshop on Semantic Evaluation (SemEval-2024), anthan. 2022. Ensemble deep learning: A review. Engineering Atul Kr. Ojha, A. Seza Doğruöz, Harish Tayyar Madabushi, Gio- Applications of Artificial Intelligence 115 (Oct. 2022), 105151. vanni Da San Martino, Sara Rosenthal, and Aiala Rosá (Eds.). https://doi.org/10.1016/j.engappai.2022.105151 Association for Computational Linguistics, Mexico City, Mex- Renhua Gu and Xiangfeng Meng. 2024. AISPACE at SemEval- ico, 403–411. https://doi.org/10.18653/v1/2024.semeval-1.63 2024 task 8: A Class-balanced Soft-voting System for Detecting Areg Mikael Sarvazyan, José Ángel González, and Marc Franco- Multi-generator Machine-generated Text. In Proceedings of the salvador. 2024. Genaios at SemEval-2024 Task 8: Detecting 18th International Workshop on Semantic Evaluation (SemEval- Machine-Generated Text by Mixing Language Model Prob- 2024), Atul Kr. Ojha, A. Seza Doğruöz, Harish Tayyar Mad- abilistic Features. In Proceedings of the 18th International abushi, Giovanni Da San Martino, Sara Rosenthal, and Aiala Workshop on Semantic Evaluation (SemEval-2024), Atul Kr. Rosá (Eds.). Association for Computational Linguistics, Mex- Ojha, A. Seza Doğruöz, Harish Tayyar Madabushi, Giovanni ico City, Mexico, 1476–1481. https://doi.org/10.18653/v1/2024. Da San Martino, Sara Rosenthal, and Aiala Rosá (Eds.). Asso- semeval-1.212 ciation for Computational Linguistics, Mexico City, Mexico, Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. 101–107. https://doi.org/10.18653/v1/2024.semeval-1.17 2021. DeBERTa: Decoding-enhanced BERT with Disentangled Areg Mikael Sarvazyan, José Ángel González, Marc Franco- Attention. In International Conference on Learning Representa- Salvador, Francisco Rangel, Berta Chulvi, and Paolo Rosso. tions. https://openreview.net/forum?id=XPZIaotutsD 2023. Overview of AuTexTification at IberLEF 2023: De- Gao Huang, Yixuan Li, Geoff Pleiss, Zhuang Liu, John E. Hopcroft, tection and Attribution of Machine-Generated Text in Mul- and Kilian Q. Weinberger. 2017. Snapshot Ensembles: Train 1, tiple Domains. Proces. del Leng. Natural 71 (2023), 275– Get M for Free. In International Conference on Learning Repre- 288. http://journal.sepln.org/sepln/ojs/ojs/index.php/pln/ sentations. https://openreview.net/forum?id=BJYwwY9ll article/view/6559 Mihaly Kiss and Gábor Berend. 2025. SzegedAI at GenAI Detec- Yuxia Wang, Jonibek Mansurov, Petar Ivanov, Jinyan Su, Artem tion Task 1: Beyond Binary - Soft-Voting Multi-Class Classifi- Shelmanov, Akim Tsvigun, Osama Mohammed Afzal, Tarek cation for Binary Machine-Generated Text Detection Across Mahmoud, Giovanni Puccetti, and Thomas Arnold. 2024. Diverse Language Models. In Proceedings of the 1stWorkshop SemEval-2024 Task 8: Multidomain, Multimodel and Multilin- on GenAI Content Detection (GenAIDetect), Firoj Alam, Preslav gual Machine-Generated Text Detection. In Proceedings of the Nakov, Nizar Habash, Iryna Gurevych, Shammur Chowdhury, 18th International Workshop on Semantic Evaluation (SemEval- Artem Shelmanov, Yuxia Wang, Ekaterina Artemova, Muc- 2024), Atul Kr. Ojha, A. Seza Doğruöz, Harish Tayyar Mad- ahid Kutlu, and George Mikros (Eds.). International Confer- abushi, Giovanni Da San Martino, Sara Rosenthal, and Aiala ence on Computational Linguistics, Abu Dhabi, UAE, 166–172. Rosá (Eds.). Association for Computational Linguistics, Mex- https://aclanthology.org/2025.genaidetect-1.15/ ico City, Mexico, 2057–2079. https://doi.org/10.18653/v1/2024. Stefan Lee, Senthil Purushwalkam, Michael Cogswell, David J. semeval-1.279 Crandall, and Dhruv Batra. 2015. Why M Heads are Better than Yuxia Wang, Artem Shelmanov, Jonibek Mansurov, Akim Tsvi- One: Training a Diverse Ensemble of Deep Networks. CoRR gun, Vladislav Mikhailov, Rui Xing, Zhuohan Xie, Jiahui abs/1511.06314 (2015). arXiv:1511.06314 http://arxiv.org/abs/ Geng, Giovanni Puccetti, Ekaterina Artemova, Jinyan Su, 1511.06314 Minh Ngoc Ta, Mervat Abassy, Kareem Elozeiri, Saad El Dine Daojun Liang, Haixia Zhang, Dongfeng Yuan, and Minggao Ahmed, Maiya Goloburda, Tarek Mahmoud, Raj Vardhan Zhang. 2025. Multi-Head Encoding for Extreme Label Classi- Tomar, Alexander Aziz, Nurkhan Laiyk, Osama Mohammed fication. IEEE Transactions on Pattern Analysis and Machine Afzal, Ryuto Koike, Masahiro Kaneko, Alham Fikri Aji, Nizar Intelligence 47, 3 (March 2025), 2199–2211. https://doi.org/10. Habash, Iryna Gurevych, and Preslav Nakov. 2025. GenAI 1109/tpami.2024.3522298 Content Detection Task 1: English and Multilingual Machine- Weixin Liang, Zachary Izzo, Yaohui Zhang, Haley Lepp, generated Text Detection: AI vs. Human. In Proceedings of the Hancheng Cao, Xuandong Zhao, Lingjiao Chen, Haotian Ye, 1st Workshop on GenAI Content Detection (GenAIDetect). Inter- Sheng Liu, Zhi Huang, Daniel Mcfarland, and James Y. Zou. national Conference on Computational Linguistics, Abu Dhabi, 2024a. Monitoring AI-Modified Content at Scale: A Case UAE. Study on the Impact of ChatGPT on AI Conference Peer Mitchell Wortsman, Gabriel Ilharco, Samir Yitzhak Gadre, Reviews. In Proceedings of the 41st International Conference Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S. Morcos, on Machine Learning (Proceedings of Machine Learning Re- Hongseok Namkoong, Ali Farhadi, Yair Carmon, Simon Ko- search, Vol. 235), Ruslan Salakhutdinov, Zico Kolter, Kather- rnblith, and Ludwig Schmidt. 2022. Model soups: averag- ine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, ing weights of multiple fine-tuned models improves accuracy and Felix Berkenkamp (Eds.). PMLR, 29575–29620. https: without increasing inference time. arXiv:2203.05482 [cs.LG] //proceedings.mlr.press/v235/liang24b.html https://arxiv.org/abs/2203.05482 47 Hybrid Reinforcement Learning Enhanced Genetic Algorithm for the Capacitated Vehicle Routing Problem with Split Deliveries and Heterogeneous Fleet ∗ † Ahmed Dabbous András Bóta Luleå University of Technology, Department of Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Computer Science, Electrical and Space Engineering, Embedded Intelligent Systems Lab Embedded Intelligent Systems Lab Luleå, Sweden Luleå, Sweden ahmdab- 1@student.ltu.se andras.bota@ltu.se Abstract algorithm, a set of genetic operators is applied to the chromo- somes, to generate the next set of solutions. Both the operators Vehicle routing is a classical field of combinatorial optimization and the algorithm itself may have a large set of hyperparameters. with a multitude of real-life applications. Recent advances in ma- Finding the one that works the best for a specific problem can be chine learning, specifically reinforcement learning, promise im- difficult. proved performance compared to traditional heuristic approaches. Reinforcement learning (also called as approximate dynamic Here, we propose a hybrid reinforcement learning enhanced ge- programming) is an interdisciplinary branch of machine learning, netic algorithm to solve a less common VRP variant: the Capac- where agents take actions in an environment aiming to maximize itated Vehicle Routing Problem with Split Deliveries and Het- a reward function. Recent advances in reinforcement learning, erogeneous Fleet. We test the performance of the approach on a such as deep Q learning, gave birth to applications in combinato- custom dataset consisting of the road network and agricultural rial optimization, including VRP [3]. buildings on the island of Gotland. Our results show, that the rein- In this proof-of-concept paper, we propose a reinforcement forcement learning algorithm is able to improve the performance learning enhanced genetic algorithm to solve the Capacitated Ve- of the genetic algorithm by guiding operator selection. hicle Routing Problem with Split Deliveries and Heterogeneous Keywords Fleet. Our solution uses a novel chromosome representation, and introduces a pair of operators on them. We use reinforce- CVRP, split deliveries, heterogeneous fleet, reinforcement learn- ment learning to dynamically learn genetic operator choice. We ing, genetic algorithm demonstrate the performance of our method on a custom dataset inspired by agricultural product collection on the island of Got- 1 Introduction land. The vehicle routing problem (VRP) is a traditional combinatorial optimization problem, with the goal of finding the optimal set 2 Problem formulation of routes for a fleet of vehicles visiting a set of locations. The Let a complete graph 𝐺 = ( ) = { } 𝑉 , 𝐷 with node set 𝑉 0, 1, . . . , 𝑛 , problem has received widespread attention from the research where node 0 denotes the depot and nodes 1, . . . , 𝑛 represent community since its first appearance. Further contributing to customers. Let 𝑑 denote the travel distance between nodes 𝑖 𝑗 𝑖 its popularity, VRP has many applications in practice and has ∈ { } and 𝑗 . Each customer 𝑗 1, . . . , 𝑛 has a non-negative demand inspired many variants [6]. A common variant, the capacitated ∈ 𝑞 . Let 𝐾 be the set of vehicles, where each vehicle 𝑘 𝐾 has a 𝑗 vehicle routing problem (CVRP) introduces a capacity constraint non-negative capacity 𝑢 . 𝑘 on the participating vehicles. The goal is to design a set of routes, each assigned to exactly Here, we focus on two less-known variants of CVRP: 1. CVRP one vehicle, such that every route starts and ends at the depot, with Split Deliveries, where multiple vehicles may serve a single and the total demand served on a route does not exceed the customer, and 2. CVRP with Heterogeneous Fleet, where a fleet capacity 𝑢 of its assigned vehicle. Multiple vehicles are allowed 𝑘 of vehicles with different capacities may serve customers. Both to visit the same node. The objective is to minimize the total of these variants model realistic situations, and as such, they are distance, defined as the sum of all travel costs over all traversed especially relevant in industry applications [14, 5]. pairs of nodes. Genetic algorithms are powerful metaheuristics frequently used to solve complex optimization tasks [1], including VRP and 3 Method its variants [12]. Genetic algorithms work with a population of Our proposed solution method is a genetic algorithm enhanced solutions, referred to as chromosomes. In each iteration of the by a reinforcement learning algorithm, which aims to optimize ∗ genetic operator selection. Both authors contributed equally to this research. † Corresponding author. 3.1 Genetic algorithm Permission to make digital or hard copies of all or part of this work for personal Genetic algorithms (GA) are powerful metaheuristics inspired or classroom use is granted without fee provided that copies are not made or by the process of natural selection [1]. They work with a set distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this (or population) of 𝑃 solutions, also called chromosomes. Each work must be honored. For all other uses, contact the owner /author(s). chromosome represents a solution to the problem. The popula- Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia tion is updated in an iterative manner, with the 𝑡-th iteration © 2025 Copyright held by the owner/author(s). being referred to as the 𝑡-th generation. The first generation 48 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Bóta et al. of solutions can be randomized, or be an output of a different on the parent chromosomes and copies the segment between method. Genetic algorithms have three main operators: 1. Se- these points from the first parent into the child. The remaining lection, 2. Crossover, and 3. Mutation. The selection operator positions in the child are filled with the elements from the second determines which solutions in the 𝑡 1-th generation progress to parent in the order they appear, with the exception of elements − the 𝑡-th generation based on the fitness value. The fitness value already present. A second offspring is created by copying over evaluates the goodness of the solution. The crossover operator the segment between the cut points from the second parent, recombines the chromosomes of two (or potentially more) parent with the remaining position filled up from the first parent. Since solutions to produce offspring solutions. The crossover should our chromosome representation allows for duplicates, repeated increase the general quality of the solutions in the population. instances of the same element are treated as identical to each The usually unary mutation operator introduces genetic diver- other. This ensures that the offsprings are valid permutations, sity within individuals to prevent premature convergence. The preserving the relative order of elements from their parents. mutation operation is intended to explore the local neighborhood We derive two crossover operators from OX1, one for the cus- of the current solution. In the rest of this section we describe the tomer orders, and one for the vehicle assignment. For customer proposed algorithm. orders (COX1) we first save the positions of vehicles in the sec- ond chromosome. Then we apply the OX1 operator only on the 3.1.1 Chromosome representation. We propose a new chromo- customer orders. Finally, in the resulting new solution, we restore some representation adapted to the specifics of heterogeneous the position of the vehicles in the exact same order they were in fleet and split-delivery problems. Our solution represents both the second chromosome. the vehicle assignment and the order of customers in the same chromosome, while allowing customers to be visited by multiple C 1 B 3 1 A 3 D 2 2 vehicles. The proposed representation is still fundamentally a permutation with repetition. Two different sets of symbols are used to represent vehicles 1 3 1 3 2 2 and customers. Here, we use letters to represent vehicles and 1 2 1 3 2 3 numbers to represent customers. Furthermore, to allow for split 1 2 3 1 2 3 deliveries, we split up the demand of each customer in the fol- B 1 2 1 A 3 C 2 D 3 lowing way: if customer 𝑗 had a demand of 𝑞 𝑗 , then the number corresponding to 𝑗 will appear 𝑞 times in the chromosome. Let-𝑗 ters representing vehicles are always followed by at most a 𝑢 B 1 2 3 A 1 C 2 D 3 𝑘 amount of numbers representing customers, where 𝑢 represents 𝑘 the capacity of the vehicle 𝑘. Figure 2: An example of applying the COX1 operator to On Figure 1, a chromosome with three unique customers and two parent chromosomes in our defined representation. four vehicles is shown. Each of the three unique customers is For brevity, only one offspring permutation is shown but a repeated in the permutation according to their demands. The second offspring is also generated by swapping the parents. vehicles in the permutation are represented by letters. Each color indicates a unique route belonging to one of the vehicles. Figure 2 provides a visual example of how the COX1 operator is applied to two parent chromosomes. The operator extracts the customer number elements from the permutation maintaining C 1 B 3 1 A 3 D 2 2 their ordering. This step generates two new permutations con- sisting solely of customer numbers. Two point OX1 crossover is Figure 1: A chromosome with 𝑛customers = 3 unique cus- applied on these new permutations to generate two new order- tomers and 𝑛 = 4 vehicles vehicles. Each color represents a ings of customers. The new customer orderings are reinserted unique vehicle route. into the parent permutations in the old customers’ positions with- out altering the vehicle positions within the parent permutations. If we consider customer 3, then based on the representation, The second operator works in a similar way, just for the vehicle the customer has 2 units of demand. The delivery of their de- assignment (VOX1). We first save the order of the customers from mands will be split between vehicles 𝐵 and 𝐴, meaning this cus- the first chromosome. Then we apply the OX1 operator only on tomer will be serviced twice in two separate deliveries. the vehicles. Then, we insert the new vehicle assignment into the customer ordering sequentially, always filling up the vehicles 3.1.2 Selection. We evaluate the fitness of the chromosomes by to their maximum capacity. calculating the sum of all route lengths. In this work, we have a fixed amount of vehicles and assume all of them have the same We use the inversion mutation operator [1]. It 3.1.4 Mutation. works by selecting two positions at random within the chromo- operating costs. The vehicles may have different capacities. The algorithm always works with a population of 100 chro- some and reversing the order of all elements between these two mosomes. We apply a limited elitism strategy with the 7 best positions. The operator preserves the permutation with repetition chromosomes automatically copied over from the previous gen- property of our chromosome representation. eration. The rest of the population is filled up with the best 93 3.1.5 Algorithm overview. Algorithm 1 summarizes the genetic solutions resulting after performing the crossover and mutation algorithm. operators. 3.1.3 Crossover. We present a modified version of the standard 3.2 Reinforcement learning two-point permutation crossover operator (OX1) [13] adapted to Reinforcement learning (RL) is a machine learning paradigm our chromosome representation. OX1 first selects two cut points where an agent learns to make decisions by interacting with an 49 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Algorithm 1 Genetic Algorithm (single-objective, generational, the quality of state action pairs by estimating the expected sum elitist) of discounted future rewards. The policy chooses the action with 1: population size 𝑃 , elite size 𝑚 , mutation rate | | Input: the maximum Q-value in each given state. During the training elites process, a Q-table containing the Q-value of each state-action 𝑝𝑚, crossover ratio 𝑝𝑐, generations 𝑇 2: 𝑃 InitializePopulation 𝑃 () ← pair is maintained and updated according to equation (1) below. 3: EvaluateFitness 𝑃 () 𝑄 new 𝑠𝑡, 𝑎𝑡 𝑄 𝑠 , 𝑎 old ( ) = ( 𝑡𝑡) 4: 𝑡 0 ← 5: + 𝛼 (𝑟𝑡 + 𝛾 argmax𝑄 (𝑠𝑡+1, 𝑎) − while 𝑄 𝑠 , 𝑎 𝑡 < 𝑇 old ( 𝑡𝑡 )) do (1) 𝑎 ★ ← ( ) 6: 𝑃 SelectElites 𝑃 , 𝑚 ⊲ copy best 𝐸 elites The learning rate 𝛼 indicates the rate at which the agent learns unchanged 7: 𝑀 ⊲ mating pool ← ∅ a new behaviour. The discount rate 𝛾 indicates the relative sig- ★ nificance of future rewards relative to the immediate reward ← ∅ 8: 𝐶 ⊲ Selected Crossover Offspring 9: 𝑀 < 𝑝 𝑃 𝑡 2 while | | 𝑐 | | 2 𝑟 . do 10: 𝑝1 SelectParent 𝑃 ⊲ Roulette Wheel Selection ( ) ← We employ a standard epsilon-greedy policy [4], where with 11: 𝑝 SelectParent 2 ← a probability 𝜖, a random action will be chosen in the given ( ) 𝑃 12: 𝑝 , 𝑝 Crossover 𝑝1, 𝑝2 ) ← () ( ′ ′ state. This allows for an exploration of the action space. With a 1 2 ′ probability of 1 −𝜖, the action argmax 𝑄 𝑠 with the maximal 𝑡 𝑎 ( ) ( ) , 𝑎 13: Append 𝑀 , 𝑝 1 14: Append 𝑀 , 𝑝 ) ( ′ Q-value given the current state in the Q-table is chosen. In our 2 15: end while setup, the 𝜖-value is initialized to 1 and is exponentially decayed ★ with every generation at a decay rate 0.0005 until it plateaus at ← ( | | − ) 16: 𝐶 SelectTop 𝑀 , 𝑃 𝑚elites 17: ★ 𝜖min 0 =.1. This allows for exploration of the action space in the 𝑐 for 𝑖 in 𝐶 do 18: 𝑐 𝑖 Mutate ← beginning when the Q-values are not well estimated. ( 𝑐𝑖 , 𝑝𝑚 ⊲ Inversion ) ★ ← 19: 𝑃 𝑐 𝑖 20: 4 Data end for 21: 𝑃 𝑃 We measure the performance of our proposed algorithm on a ← ★ 22: EvaluateFitness 𝑃 novel custom dataset based on open access data on agricultural ( ) 23: 𝑡 𝑡 1 buildings on the island of Gotland in Sweden. The dataset was ← + 24: originally inspired by livestock collection problem [9] but we end while 25: Best 𝑃 use it in a different context in this paper. All data were down-return ( ) loaded from OpenStreetMaps https://www.openstreetmap.org/. The road network is the road network of Gotland. A 𝑛 250 = number of delivery locations were sampled from all locations environment to maximize a reward function. The agent observes marked as agricultural in the OpenStreetMaps database. Pairwise a state, takes an action to move to a new state, and potentially distances between customer nodes were computed with the Open- receives a reward, with the overall goal of finding a policy, map- SourceStreetMap library https://www.openstreetmap.org while ping from states to actions, that maximizes expected discounted the customer-depot distances were computed with the Geopy rewards. RL can be combined with genetic algorithms to create open source library https://geopy.readthedocs.io/en/stable/. hybrid optimization methods combining the strengths of both The demands for each customer were sampled from a uniform paradigms. [8, 10, 11] [ ] distribution in the interval 1, 10 while the vehicle capacities In this paper, we use Q-learning [4], a model-free value-based were likewise sampled from a uniform distribution in the inter- approach to enhance our genetic algorithm, inspired by the works [ ] val 100, 200 . The depot position was chosen to be the mean of [8, 11]. longitude and latitude of the customer positions. The state space is defined by a pair of categorical values, which categorize the percentage change in the fitness value of the pop- 5 Results ulation’s best member, and the diversity within the entire popu- In this proof-of-concept work, our main goal is to measure the lation [7] (the number of unique individuals in the population performance improvement reinforcement learning can provide divided by the total population size). The best fitness percentage for the genetic algorithm. We also conduct a limited amount of changes between generations are discretized into one of six cate- hyperparameter optimization for both the reinforcement learning gories ranging from high change to no change at all. Likewise, and genetic algorithms. the diversity ratio is binned into one of five categories ranging Starting with the latter, we found that the following values from very high to very low diversity. provided the best fitness population in our experiments: The action space is composed of 8 possible actions where each action is a triplet of ’C’ or ’V’ markers indicating whether a VOX1 • | | = population size 𝑃 100 or COX1 crossover operator should be used. For example, the 𝑐 𝑝 • crossover rate = 1 action triplet ’CVV’ indicates that the COX1 operator is to be 𝑚 mutation rate • 𝑝 = 0.10 used on the first generation, followed by the VOX1 operator on elites • = number of elites 𝑚 7 the second and third generations. Given that these action triplets • learning rate 𝛼 = 0.3 apply to three consecutive generations, all model training steps • = discount rate 𝛾 0.7 including sampling the actions, states, and rewards as well as • 𝜖-greedy strategy as described in section 3.2 updating the Q table are only applied on every third generation. To measure the performance improvement of the RL enhanced The Q-learning agent aims to learn the optimal state-action algorithm, we ran experiments on the Gotland dataset. To provide value (Q-value) for each state-action pair. The Q-value measures a baseline, we used the genetic algorithm described in section 50 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Bóta et al. 3.1 without any RL component, and where the two operators for both the reinforcement learning and genetic algorithms to VOX1 and COX1 were alternating in subsequent generations. We find the values providing the best performance. We will also apply compared the performance of the baseline algorithm with the our algorithm on other datasets, such as the Augerat benchmark RL enhanced version, which aims to learn the optimal operator [2]. triplet as described in section 3.2. Figure 3 shows the fitness value We also plan to extend both parts of the algorithm. So far we of the populations of both variants of the algorithm for the first have only experimented with the two genetic operators described 10000 generations. in section 3.1. We plan to conduct experiments with both tradi- tional operators, and potentially new ones. The reinforcement learning algorithm can be improved in multiple ways. It can be used to dynamically optimize the hyperparameters of the genetic algorithm during the evolution process, such as the number of elites and, the crossover and mutation rates. We can also exper- iment with more complex operator strategies, possibly adding more operators to the mix. Finally, in the context of deep Q learn- ing, neural networks can be used to learn the hyperparameters of the operators, such as the cut points of OX1 or even a binary mask similar to the one used in the PBX operator [13]. These modification will likely further enhance the performance of our algorithm. Acknowledgements This work was supported by Sweden’s Innovation Agency, grant number 2023-02595, which is gratefully acknowledged. References Figure 3: Fitness values for both baseline and RL enhanced [1] Bushra Alhijawi and Arafat Awajan. 2024. Genetic algorithms: theory, ge-algorithms on the Gotland dataset for the first 10000 gen- netic operators, solutions, and applications. Evolutionary Intelligence, 17, 3, erations. 1245–1256. [2] Philippe Augerat, D Naddef, JM Belenguer, E Benavent, A Corberan, and Giovanni Rinaldi. 1995. Computational results with a branch and cut code Our results show, that the RL-enhanced algorithm maintains for the capacitated vehicle routing problem. [3] Ahmad Bdeir, Simon Boeder, Tim Dernedde, Kirill Tkachuk, Jonas K Falkner, an advantage in allowing the agent to decide the genetic crossover and Lars Schmidt-Thieme. 2021. Rp-dqn: an application of q-learning to operator triplets throughout the experiment. Up to the 1000𝑡 ℎ vehicle routing problems. In German conference on artificial intelligence generation, both approaches follow approximately equal trajec- . Springer, 3–16. (Künstliche Intelligenz) [4] Jesse Clifton and Eric Laber. 2020. Q-learning: theory and applications. tories, attributed to the epsilon value 𝜖 > 0.6 still being relatively Annual Review of Statistics and Its Application, 7, 1, 279–301. high. This leaves the agent in an exploratory phase where it [5] Simos Efthymiadis, Nikolaos Liapis, and George Nenes. 2023. Solving a heterogeneous fleet multi-compartment vehicle routing problem: a case is more likely to choose random actions from the action space. study. , 10, 1, International Journal of Systems Science: Operations & Logistics During the evolution process, the RL agent picks up an advan- 2190474. tage as it begins to learn the crossover triplet selection. At the [6] Burak Eksioglu, Arif Volkan Vural, and Arnold Reisman. 2009. The vehicle routing problem: a taxonomic review. , Computers & Industrial Engineering end, the best fitness values of the baseline and RL-enhanced al- 57, 4, 1472–1483. gorithms were 2616 km and 1671 km respectively, showing an [7] Nguyen Thi Hien and Nguyen Xuan Hoai. 2006. A brief overview of popu-improvement of 56.5 percent. lation diversity measures in genetic programming. In Proc. 3rd asian-pacific workshop on genetic programming, hanoi, vietnam, 128–139. Further enhancing the usefulness of the algorithm the run- [8] Eda Köksal Ahmed, Zengxiang Li, Bharadwaj Veeravalli, and Shen Ren. times for the baseline and RL-enhanced algorithms were 45.1 2021. Reinforcement learning-enabled genetic algorithm for school bus scheduling. , 26, 3, 269–283. doi: Journal of Intelligent Transportation Systems minutes and 42.5 minutes respectively, showing only a slight https://doi.org/10.1080/15472450.2020.1852082. increase for the enhanced variant. This is due to the computa- [9] Johan Oppen and Arne Løkketangen. 2008. A tabu search approach for tional overhead introduced by the RL algorithm, but we believe the livestock collection problem. , 35, 10, Computers & Operations Research 3213–3229. it is compensated by the improvement in the quality of the final [10] Jose Quevedo, Marwan Abdelatti, Farhad Imani, and Manbir Sodhi. 2021. solution. Using reinforcement learning for tuning genetic algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference Companion(GECCO 6 Conclusions and future work ’21). Association for Computing Machinery, Lille, France, 1503–1507. isbn: 9781450383516. doi: 10.1145/3449726.3463203. In this paper, we proposed a reinforcement learning enhanced ge- [11] Jose Quevedo, Marwan Abdelatti, Farhad Imani, and Manbir Sodhi. 2021. Using reinforcement learning for tuning genetic algorithms. In Proceedings netic algorithm to solve the Capacitated Vehicle Routing Problem of the genetic and evolutionary computation conference companion , 1503– with Split Deliveries and Heterogeneous Fleet, and measured its 1507. [12] A Serdar Tasan and Mitsuo Gen. 2012. A genetic algorithm based approach performance on a custom dataset inspired by agricultural product to vehicle routing problem with simultaneous pick-up and deliveries. Com- collection on the island of Gotland. Our initial results show a puters & Industrial Engineering, 62, 3, 755–761. significant, 56.5 percent performance boost of the RL-enhanced [13] Anant J Umbarkar and Pranali D Sheth. 2015. Crossover operators in genetic algorithms: a review. , 6, 1. ICTACT journal on soft computing algorithm compared to a baseline genetic algorithm, that does not [14] Jiao Zhao, Hongxia Dong, and Ning Wang. 2023. Green split multiple- have a learning component. Furthermore, due to the lightweight commodity pickup and delivery vehicle routing problem. Computers & nature of the learning algorithm, the performance boost comes , 159, 106318. Operations Research with only a slight computational overhead. We plan to perform a more thorough testing of the algorithm in the future. First, we plan to perform hyperparameter optimization 51 Empiric results on the achievable performance gains by the inclusion of instance-specific information in a scheduling optimizer Mate Hegyhati hegyhati.work@gmail.com University of Sopron Sopron, Hungary University of Pannonia Veszprem, Hungary University of Applied Sciences Wiener Neustadt Wiener Neustadt, Austria Figure 1: Proposed workflow with instance-specific binaries Abstract simulation/optimization software. In each of the mentioned It is common in computational combinatorics, graph theory, cases, the software executes generic logic on the data that and combinatorial optimization to develop custom algo- is read from a database, file or received over the network, rithms. These algorithms usually address a whole problem etc. The software itself is not specific to each use-case, class, i.e., an infinitely large set of instances. As a result, which is obviously a big benefit in reusability, saving costs, their implementation is also generic, and instance-specific and a reason, why the industry developed in this direction. information is only available in the execution phase of Moreover, this phenomenon usually extends to multiple the solver, not at its compilation. There are reasonable layers. For example: logic on a website is usually written in arguments, why earlier inclusion of instance-specific infor- JavaScript, which is a (mostly) interpreted language, i.e., mation may result in better performance, i.e., compiling an even more generic code (JavaScript engine) is run, that an instance-specific solver rather than relying on a generic reads the website code, whose execution reads the JSON binary. The goal of this work is to investigate the achiev- retrieved from the backend. able performance gains on the C++ implementation of an However, rarely changing parameters and a need for effi- S-graph based makespan-minimization algorithm that tar- ciency can sometimes lead to code that has problem-specific gets job-shop problems. Results show that instance-specific data baked in for faster execution that ultimately leads to binaries can significantly outperform the generic solver, reduced code, better user experience, etc. Such examples reducing the computational needs by 70-80%. are easier found in the early days of computing, where resources were expensive and more often used for a single Keywords purpose. To name an example, timetabling programs some- optimization, scheduling, instance-specific implementation times had railway network related information hardcoded in them[7]. Today, it is rather rare to see problem-specific 1 Introduction binaries, though one could argue for example, that G-codes are problem-specific instructions on a CNC machine. In It is rather natural to have a clear distinction and separa- some very edge-cases, where performance is of utmost im- tion in our mind between a computer program and the data portance, baked-in data can be present, such as aerospace it uses in a certain scenario. This is true for almost all soft- or avionics firmware, or a missile guidance computer. It is ware, let it be a spreadsheet application, restaurant menu also common in the automotive industry to heavily rely on page on a website, training advisor on a sport watch, or a code-preprocessing macros, and generate specific binaries. Instance-specific precompiled data appears more often, e.g., MATCOS 2025, 9–10 October 2025, Koper, Slovenia BSP data and PVS in video games[9] or models in Finite 2025. 52 MATCOS 2025, 9–10 October 2025, Koper, Slovenia Mate Hegyhati Element Analysis[1], and one may even consider neural 3 Problem definition & solution nearly philosophical detour, optimization is a computation Returning to the main topic of this paper from this The scheduling algorithms of the S-graph framework[6] network weights as precompiled data as well. algorithm is taken into account to parametrize the execution of the processing times. The objective is to find the schedule with solution software, which has a similar motivation to that minimal makespan. of this paper, but differs in the approach. For the description of the algorithm, the terminology of This paper presents the results of an experiment about batch process scheduling is used instead of shop scheduling, the performance consequences of using instance-specific as it fits to an S-graph based approach better. The model is information compiled into a graph-based scheduling solver a directed graph, where each task and product (delivery) is as shown in Figure 1. represented by a vertex, and timing differences between the In Section 2 a brief technical background is given, why start of their execution are expressed by directed weighted an instance-specific binary may have performance benefits. arcs. In a fully scheduled graph, the longest path equals to after, similar to the aforementioned simulation examples. In- for the sake of keeping the descriptions brief, the problem stance Specific Parameter Tuning Strategies are researched class was reduced to job-shop scheduling with arbitrary for heuristic approaches[2], where instance-specific data job lengths, machine revisits, and non-negative integer heavy area, where every reduction in cost is highly sought their graph model and custom branching rules. However, are great candidates for the planned experiment due to Section 3 provides the definition of the problem class for the the makespan, and it also provides the bounding function experiment and the algorithm used by the implementation for partially scheduled cases. At the root of the B&B tree, described in Section 4. Finally, Section 5 shows the test the processing queue of each unit is empty, and only the results, and conclusions are drawn in Section 6. arcs expressing the production sequence of products are included. In each branching step, a unit with non-complete queue is selected, and the next task is fixed for each child 2 Brief technical background node, by inserting an arc between the previous task and the next task with the weight of the processing time of While big-O notation is an excellent tool to compare the the previous task. If a cycle in the graph is created, the theoretical efficiency of algorithms, in practice, the actual schedule is infeasible. running time matters the most. However, even on the same This is similar to the so-called EQ-based approach of the machine with the same operating system, the CPU time S-graph framework, however, it does not specify, which par- depends not only on the algorithm’s design, but on how it tial schedule in the B&B tree is investigated next, and how is expressed in the computers language. Implementations the unit is selected for branching. Fine tuning these strate- of the same algorithm can vary in many ways, which can gies can also have a significant effect on performance [5]. also lead to very significant performance differences.[3] It is To remove this factor from the comparisons, the following a well known fact in software development, that lower level strategy is used: languages such as C, C++, Rust, etc. generally outperform Partial schedules are explored in a depth-first-search • interpreted languages (JavaScript, Python) or languages manner. that are generally compiled to a virtual machine (Java, At each, the processing unit with smallest index is • C#). Most of the performance difference can be attributed selected among those that have an unfinished pro- to heap vs. stack allocations, advantages of static typing, duction queue. cache locality, and various compiler-backend optimizations. These factors can also lead to very different results for two As a result, the top of the B&B tree contains decisions implementations in the same language. related to unit 0, the levels below that to unit 1, etc. This In this study, C++, a well-established language[8] is used strategy allows a simple recursive implementation, with for testing the possible performance benefits of instance- the following logic: ery aspect writing efficient C++ code[4] here, one key factor subroutine EXPLORE( case , unit , prev , curr ) : specific implementations. While it is impossible to detail ev- is worth highlighting. The algorithm described in the next global best Section iterates over many partial schedules, whose data case = case . copy () is mostly encoded by a 2-dimensional matrix of weights, i f prev != NULL: that is copied from a parent node and altered in the child i f case . has_path ( curr , prev ) : return nodes. case . add_arc ( prev , curr , prev . proctime ) A significant difference between a generic and an instance- i f case . longest_path () >= best : return compilation time in the latter case, which allows much best = case . longest_path () faster copying. While basically one memcpy call is enough else specific solver is, that the size of this matrix is known at i f case . is_complete ( ) : to copy an arraySIZE> : on i f case . is_unit_complete ( unit ) : , doing the same allocations, and a syscall if the heap allocated to the pro- unit = case . next_incomplete_unit () vector> usually results in SIZE+1 heap cess is insufficient. Moreover, things like tasks assigned to for task in unit . tasks ( ) : the same unit do not need to be looked up dynamically, EXPLORE( case , unit ,NULL, task ) they can be hardcoded into the binary, which may also else : result in faster execution. for task in unit . tasks ( ) : 53 Performance of an instance-specific scheduling optimizer MATCOS 2025, 9–10 October 2025, Koper, Slovenia i f not case . is_scheduled ( task ) : very different binaries based on the compiler used, and EXPLORE( case , unit , curr , task ) the applied configurations. All binaries used were com- All implementations in Section 4 rely on this same ex- piled with g++ (GCC version 14.2). After testing various acceleration techniques of the S-graph framework are not optimization level 1, and another with optimization level utilized in any of the implementations. This adjusted S- 3. Initial tests showed, that -O2 and -Ofast did not show graph algorithm may not be the most efficient job-shop ploration logic of the search space. Further algorithmic binaries were generated for each implementation: one with compiler configurations on a smaller set of examples, two scheduler, however, it still retains the same important char- significant difference compared tot-O3, and naturally, no acteristics, as the general solvers for the wider problem instances were tested with all of the 5 implementations and optimization always provided worse results. Altogether 80 class with precedential recipes, mixed intermadiate storage 2 compiling options. The size of the instances ranged from policies, changeover times, etc. 3 to 8 products with 2-6 stages carried out by 5-8 units. 4 Generic and instance-specific Overheads implementations Instance-specific solvers have an overhead for generating A shared feature of all of the S-graph implementations is and compiling the source files. This is especially heavy for to maintain a longest path matrix with a cached maximum I3 . The largest such overhead was less than 6 seconds for instead of the schedule-graph itself. This removes the need I3 with-O3 , for a problem, whose solution was measured for dynamic longest path calculations and cycle checks. in hours. For the same problem the overhad in case of Maintaining such a matrix when inserting an arc takes I2 with-O1 was less then 1 second. On the other hand, 2 O ( | V | ) time in the worst case, however, this technique constructing the root node is faster for the instance-specific was proven to be advantageous over the years. The recipe binaries, but that only milliseconds and microseconds in is always stored and accessible as constant global data. the worst case for the generic and instance-specific solvers, Altogether, there are 5 implementations that will be respectively. As a conclusion, these overheads did not play compared in the next section. a significant role, except for small problems, where the G1 is the generic implementation, where a partial sched-optimal solution can be found in a few seconds either way. ule is stored in such a structure: struct Optimization levels partial_schedule { vector> longest_paths ; A somewhat unexpected result between the-O1 and-O3 uint optimization option for the 5 implementations is shown on longest_path_max ; vector a box-plot in Figure 2. < vector < uint >> unscheduled_tasks ; }; In G2, the list of unscheduled tasks for each unit is not stored by a vector in this struct, but as a managed global boolean array, which is possible due to the DFS- like strategy. Comparing these two solvers will show, how much time is needed for managing several relatively short dynamic arrays. The implementations I1 and I2 are the problem scpecific equivalents, where fix-sized arrays are employed instead of vectors, and the subroutine responsible for returning the root partial_schedule is constexpr with the return of a list-initialized structure instead of opening the input file, and reading the data dynamically. I3 is a special variant of I2, where the branching subroutine is instantiated for all possible unit, prev, curr triplets in compile time, thus, some of the conditionals and loops can be removed in exchange for longer compilation time and a larger binary. It is important to note, that all implementations traverse exactly the same DFS-like B&B tree with the same number of nodes, copies, etc. There are no algorithmic differences between them in this sense, and the C++ code for exploring Figure 2: CPU time ratio of binaries with-O3 and a node is the same in G1 and I1 , and similarly the same-O1 for the 5 implementations in the remaining three. 5 Empiric results The figure shows the distribution of CPU time with-O3 divided by that of with -O1 of all the test cases for all Test cases implementations. While several outliers are cropped from All of the test cases were generated randomly and solved the diagram, it is clear to see, that in case of the general by all of the aforementioned implementations on the same solvers, more optimization had minor benefits, but the machine. A single C++ source file, however, can result instance-specific solvers performed significantly better with 54 MATCOS 2025, 9–10 October 2025, Koper, Slovenia Mate Hegyhati less optimization. Thus, in later comparisons results from Table 1: Overall results for some examples the general solvers always refer to the-O3 build, and to the-O1 build for instance-specific solvers. case # tasks G2-O3 I2-O1 Reduction 66 40 11180 s 3246 s 71% The overall performance of 45 30 62 s 14 s 77% I2 and I3 were very close 0 38 40 s 9 s 77% for all of the larger test cases. For the 5 largest ones, Instantiated functions 24 36 7392 s 2107 s 71% I2 1 39 19 s 4 s 79% outperformed I3 by 1,1,0,9,3 percent. I3 is removed 17 33 12 s 3 s 72% from further discussion, as function instantiation did not 69 37 8 s 2 s 78% bring measurable results for this problem class, only slight performance decrease on top of the compilation overhead. Unscheduled task set representation is reasonable to assume, that syscalls for more heap space did not play a significant role for the larger examples. Figure 3 shows the effect of storing the set of unscheduled The last column of the table shows the reduction in CPU tasks differently in case of the generic and instance-specific time by using insta-specific solvers, which are all in the solvers. 70-80% range. 6 Concluding remarks While computer programs are meant to be generic for many reasons, performance focused applications may ben- efit from generating binaries that have instance-specific information encoded. In this work, this question was put to the test on an S-graph-based custom scheduler for job-shop problems, and results showed that a 70% reduction in com- putational needs is achievable mostly by fixing container sizes in compile time. This initial investigation motivates more exhaustive experiments on more complex real-life case-studies. References [1] Klaus-Jürgen Bathe. 2002. Finite Element Procedures. Prentice- Hall of India. isbn: 8120310756. http://www.worldcat.org/isbn /8120310756. [2] Yasemin Eryoldas and Alptekin Durmuşoğlu. 2021. A literature survey on instance specific algorithm configuration methods. Proceedings of the International Conference on Industrial Engineering and Operations Management. https://api.seman ticscholar.org/CorpusID:240158129. Figure 3: Effect of different unscheduled task set [3] M. Hegyhati and Zs. Tuza. 2023. Why implementation matters for graph algorithms? Presented at OPAL, Veszprém, Hungary. management (June 2023). [4] Scott Meyers. 2005. Effective C++: 55 Specific Ways to Improve Your Programs and Designs. (3rd ed.). Addison- The Figure shows an XY plot, where the X and Y coordi- Wesley Professional. [5] Zsolt Ádám Nemes. 2015. Heuristic Accelerations of the S- For most of the instances, both coordinates are less than 1, [6] E. Sanmartí, Luis Puigjaner, Tibor Holczinger, and Ferenc Friedler. 2002. Combinatorial framework for effective schedul- the exceptions belong to smaller instances, i.e., maintaining nates are CPU time ratios G2/G1, and I2/I1, respectively. graph Solver. Bachelor’s thesis. University of Pannonia. ing of multipurpose batch plants. AIChE Journal, 48, 2557– a single boolean array was more beneficial, as expected. 2570, 11. doi:10.1002/aic.690481115. It can also be observed, that in general, this had a larger [7] G. R. Stibbs. 1958. Railway timetabling and early computers. effect on instance-specific solvers, whose partial problem Proceedings of the IEE. [8] Bjarne Stroustrup. 2013. The C++ Programming Language. representation would not use vectors for other purpose. (4th ed.). Addison–Wesley, Boston, MA. isbn: 978-0321958327. [9] Seth J. Teller and Carlo H. Séquin. 1991. Visibility prepro- Largest instances cessing for interactive walkthroughs. SIGGRAPH Comput. Graph., 25, 4, (July 1991), 61–70. doi:10.1145/127719.122725. Table 1 shows the overall results for several examples, including the largest cases and some medium-sized ones. The second column shows the number of tasks, which is the maximal depth of the B&B tree, thus the maximal number of function calls on the stack. Memory usage grows cubically with this in these DFS-like implementations. As is shown in the table, CPU becomes an issue much faster than RAM. Moreover, considering the 4 bytes for an unsigned int, the I2 solver takes 256 kB to store all weights in the matrices in the call stack when a complete schedule is examined. This number is definitely higher for G2, but it 55 Bi-Level Routing and Scheduling Alain Quilliot and H´el`ene Toussaint LIMOS Lab. UCA, CNRS and EMSE Clermont-Ferrand, France Email: alain.quilliot@uca.fr Abstract—Decentralized renewable energy platforms promote when the vehicles return to PVP to recharge. We suppose that self-consumption, a single agent becoming able to simultaneously the system is deterministic . Its main components are: act as a producer and a consumer of power. We deal here with the design of routing strategies for electric vehicles relying on a local photovoltaic (PV) platform. We propose an exact Mixed-Integer • A Photo-Voltaic platform PVP Linear Programming (MILP) setting solved through Branch and The time horizon of PVP is divided into N periods Cut, along with a heuristic approach involving an approximation i = 0, . . . , N − 1, all with a same duration p. Thus of the behavior of the PV platform. period i starts at time p · i and ends at time p · (i + 1). During this period, PVP is expected to produce R i power units. It may also buy an additional amount yi of I. INTRODUCTION Ch power, that cannot exceed a threshold C . The cost of Renewable energy sources are promoting the emergence yi may be written Φi(y), where Φi is a piecewise linear of the increasing convex function, convexity meaning that self-consumption paradigm [6], according to which consumers become energy producers. This paradigm raises its marginal power purchase prices are usually increasing. own issues, operational ones (synchronization of production PVP is provided with a macro-battery, with storage and consumption) and tactical (pricing). 0 capacity P V P Ch P V P C ≥ C and initial load H . It must We consider here a meet vehicle’s demand, ending with a charge at least Vehicle Routing under Energy Production Constraints equal to P V P H under minimal purchase cost P Cost. VR EPC problem which consists in routing 0 electric vehicles (EV) while synchronizing their activity with the time dependent production and purchase of power. • An Electric Vehicle EV and a Set of Customers J Numerous routing models (see EV is initially located at a Depot. It must visit, within Green Vehicle Routing , Pollution Routing the time horizon [0, N.p], a set J = {1, . . . , M} of models [2], [7], [9],[3] ) have been designed, involving refueling transactions subject to time customers according to a TSP (Traveling Salesman) windows or shared access constraints, as well as CO2 route Γ, before coming back to Depot. Moving from a emissions (see Kuo [4], Lajunen [5]). Nevertheless, few customer j to a customer k requires Ej,k energy units studies simultaneously dealt with both energy production and Tj,k time units. Vectors T and E define distances on and consumption, which requires integrating heterogeneous the set defined by Depot, PVP and J, Depot and PVP routing and scheduling processes (see [1]). Related models being considered as customers identified as Depot = 0 are complex ones, involving highly heterogeneous variables at the beginning of the process, Depot = M + 1 at the and relying on the existence of a single central decider, which end and PVP = −1. EV is provided with a battery, with may not fit most real life contexts. In any case, designing storage capacity EV EV C and initial load H . It must end 0 efficient algorithms remains a challenge. with a charge at least equal to EV H while minimizing 0 the time EV T ime when it is back to Depot. Because of the collaborative features of our problem, we adopt here the point of view of the EV and propose: • Recharge Events EV must periodically move to PVP in order to recharge. • An exact MILP model that considers the vehicle as A recharge event requires a single period i and takes leader and includes Recharge Decomposition constraints place between two customers j, k consecutive according requiring the application of a separation time-polynomial procedure. We solve it through branch-and-cut. Ch EV an amount to Γ: EV moves from j to PVP before time p · i, receives m ≤ Inf (C , C ) of power, and leaves • Heuristic algorithms, that approximate the behavior of the PVP at time p · (i + 1). This recharge event, denoted power producer. by ω = (i, j, k, m), induces a cost Ψi which depends on i and is independent on m (infrastructure resource II. THE VR EPC P ROBLEM cost). Purchasing power is forbidden during the period i. We consider a photovoltaic (PV) platform, referred to as PVP, Recharge event ω may force EV to wait at PVP until the along with an electric vehicle (EV) in charge of visiting a set of beginning of period i in case it arrives before time p · i. customers. These two players interact through recharge events If we denote by τ j the time when EV arrives at j, by 56 V EV P V P P V P P V P its energy load and by V the energy load of – Non negative L , i = 0 , . . . , N − 1 : L is the j i i i the PV-plant at the beginning of i, then: power transferred at i if δ i = 1. – p · i ≥ τ j + Tj,−1; τk = p · (i + 1) + T−1,k; • Recharge Event variables: – P V P P V = V − m ≥ 0; i+1 i – {0, 1}-valued U , i = 0, . . . , N − 1, j = 0, . . . , M: i,j – EV The load of EV at the end of i is equal to V j U − = 1 means that some recharge event involving i,j E EV j, −1 j + m ≤ C ; occurs at period i; – The load of EV at the beginning of i is equal to – Non negative m i,j, i = 0, . . . , N − 1, j = 0, . . . , M: V EV − E j j,−1 0 ≥. m means related amount of power. i,j Given time versus money coefficient α, we set VR EPC as a A. Structural EV Constraints bi-level mono-objective problem, with EV a leader. Z and X describe the full route followed by EV. They must VR EPC: Vehicle Routing under Energy Production Con- clearly meet the standard vehicle routing constraints: straints: {Compute the route Γ followed by EV together with • ZM +1,0 = 1; ∀j : Zj,j = 0; (VR1) the recharge events linking P PVP and EV in such a way that: • ∀ j = 0 , . . . , M + 1 : Z = 1 = k=−1,...,M+1 j,k • P Z All customers are visited once within the time horizon. ; (VR2) k=−1,...,M +1 k,j • P P Vehicle storage capacity constraints and energy require-• Z = Z j=0,...,M +1 j, −1 1 ≥; (VR3) j=0,...,M+1 −1,j ments are satisfied. • ∀j, k ∈ {0, . . . , M + 1}: X j,k ≥ Zj,k ; (VR4) • Extended cost P Cost+α.EV T ime is minimized, under • ∀j ∈ { P 0 , . . . , M + 1 } : X = 1 = k =0 , 1 ,...,M +1 j,k the P PVP constraints: X ; (VR5) k k,j =0 , 1 ,...,M +1 – y meets the demand related the recharge events; – It meets PVP storage and charge capacities as well (VR1, ..., VR5) do not ensure that Z, X defines a route, as the constraint about the final charge of PVP; consistent with the power requirements. We reinforce them by – P P Cost =(Ψ noticing that if EV spends W energy inside or at the border · δ + Φ ( y )) . } i i i i i of some customer subset A which does contain PVP, then it must move at least W ⌈ ⌉ times towards PVP in order to III. A MILP M CEV ODEL S OLVED THROUGH B RANCH &C UT recharge. Let us set: Since we adopt here the point of view of the vehicle, our • For any such a subset A of {−1, 0, . . . , M, M + 1}: master variables are 2 {0, 1}-valued vectors Z = (Zj,k, k ̸= – Cl(A) = {(j, k) s.t at least j or k is in A}; j ∈ {− 1 , 0 , . . . , M + 1 } ) and X = ( X j,k , k ̸ = j ∈ – δ ( A ) = { ( j, k ) , s.t j / ∈ A and k ∈ A } . { 0 , . . . , M +1 } ) , describing the routes followed by the vehicle with and without the recharge detours respectively. Those main • For any (j, k): EV – EV Π variables are completed by time and charge EV variables, = E if ( j, k ) = ( M + 1 , 0) and Π = C j,k 0 j,k by else. PVP variables and by Recharge Event variables. • EV variables: ∗ EV Π – ∗ EV Π = C − E (j, k M + 1, and j,k 0 if ) = ( 0) – {0, 1}-valued Zj,k j,k = C else. , k ̸ = j ∈ {− 1 , 0 , . . . , M + 1 } : Zj,k = 1 iff EV moves from j to k: (-1 = PVP). – Then we derive the Recharge Decomposition constraints: { 0 , 1 } -valued X j,k , k ̸ = j ∈ { 0 , . . . , M +1 } : X j,k = 1 P iff EV moves either from j to k or from j to PVP , • For any A ⊆ { 0 , . . . , M + 1 } , Π · Z ≥ ( j,k j,k j,k ) ∈ δ ( A ) and next from P PVP to k . E · Z (VR6) (j,k) ∈Cl(A) j,k j,k – EV M P P ∗ Non negative L , j = 0 , . . . , M : L is the power • For any A ⊆ { 0 , . . . , M +1 } , Π ·Z ≥ j j (j,k)∈δ(J j,k − A ) j,k transferred to P EV just after j if Z j, −1 j,k = 1 ; E · Z (VR6-Bis) ( j,k j,k ) ∈ Cl ( A ) – EV EV Non negative V , j = 0 , . . . , M + 1 : V is the j j power stored by EV when it arrives at j; Lemma 1: The VRP constraints (VR1, ..., VR6-Bis) hold if and – Non negative τj, j = 0, . . . , M + 1: τj is the time only if the arcs (j, k) such that Z j,k = 1 define a collection when EV arrives at j; γ of sub-tours γs, s = 0, . . . , S such that: – ∗ ∗ Non negative τ , j = 0 , . . . , M + 1 τ j j • γ : is the time 0 starts from Depot = 0, ends into PVP = −1, and when EV starts recharging after j if Z EV j, − 1 = 1 . spends less than H power. (SUB1) 0 • PVP variables: • γS starts from PVP = −1, ends into Depot = 0, and – EV EV Non negative y , i = 0 , . . . , N − 1 : y means the spends less than C – H power. (SUB2) i i 0 power bought by PVP during period i; • For any s = 1, . . . , S −1, γs starts from PVP = −1, ends – EV { 0 , 1 }-valued δ , i = 0 , . . . , N − 1 : δ = 1 means into − 1 and does not require more than C power. i i that some recharge event takes place at i; (SUB3) – P V P P V P Non negative V , i = 0 , . . . , N 1 , N i • − : V is Every customer j = 1, . . . , M is visited once. (SUB4) i the charge of PVP at the beginning of i; 57 Remark 1: Constraints (VR6, VR6-Bis) do not order the TABLE I sub-routes BEHAVIOR OF VR EPC MILP γ s , s = 0 , . . . , S . Id (N, M, p) LBG LP U BG LP Separating the Recharge Decomposition Constraints: 3 (160, 10, 1) 51,6 171 Given 2 possibly non integral vectors 3-2 (160, 10, 2) 51,7 174 ( Z, X ) , separating the constraints (VR6, VR6-Bis) means checking that all 5 (240, 20, 1) 43,8 148 6 (240, 20, 1) 59,5 176 those constraints are satisfied by (Z, X ) and, in case they 5-2 (240, 20, 2) 43,9 148 are not, computing a contradicting subset A ⊆ {0, . . . , M +1}. 6-2 (240, 20, 2) 59,1 201 8 (320, 30, 1) 58,0 151 Theorem 1: The Recharge Decomposition constraints can 8-2 9 (320, 30, 1) 51,7 117 (320, 30, 2) 58,0 153 be separated in polynomial time, via a min cost flow algorithm. 9-2 (320, 30, 2) 51,2 127 Principle of the Proof: We construct an auxiliary multi-graph G EV = ( X, A ) with X = { Source = − 1 , 0 , . . . , M +1 , M +2 = – ∀ j = 0 , . . . , M : Z j, −1 j ( → V ≥ E ; (VR10) ) j, −1 Sink EV EV } and A is defined as the set of all simple-arc ( j, k ) , j ̸ = – ∀ j = 0 , . . . , M : Z j, −1 ( → V + L ≤ j j k EV ∈ {− 1 , 0 , . . . , M + 1 } , augmented, for every j , with copy-E + C ) ; (VR10-Bis) j,−1 arcs k ( j, M + 1), k ̸= j ∈ {−1, 0, . . . , M + 1}, connecting j – τ = 0; τ ≤ p · N; (VR11) 0 M +1 to Sink = (M + 2) and provided with label k. Every copy- – ∀j, k = 0, . . . , M + 1: Zj,k → arc k a = ( j, M + 1) is provided with a weight w equal to (τ + T ≤ τ ); (VR12) a j j,k k Ej,k · Zj,k. Every simple-arc a = (j, k) is provided with a – ∀j, k = 0, . . . , M + 1: (Xj,k − Zj,k = 1) → weight ∗ w a j,k j,k j,k j,k j ( equal to Π · Z − E · Z . Then computing τ + p + T ≤ τ −1,k k ); (VR13) A that contradicts VR6 (VR6-Bis) means computing some cut – ∀j = 0, . . . , M + 1: (Zj,−1) → B ∗ which separates Source = − 1 from Sink = M + 2 in G ( τ j j, − + T 1 j τ ≤ −); (VR13-Bis) and is such that P w does a s.t. origin ′ ′ a ( a ) ∈ B ,destination ( a ) ∈ / B not exceed some threshold. This can be done in polynomial • Synchronization Constraints time through a Max-Flow algorithm. End-Proof. P – ∀j = 0, . . . , M : U = Z i =0,...,N i,j j, − 1−1 ; (SY1) B. The VR EPC MILP MILP formulation – ∀j = 0, . . . , M : We get it while distinguishing 3 main groups of constraints: ,...,N −1 j P – P EV m i=0 i,j = L ; (SY1-Bis) • ∀i = 0, . . . , N − 1: U = δ ; (SY2) j =0 ,...,M i,j i The PVP Constraints : They involve the variables related – ∀ i = 0 , . . . , N − 1 : to the purchase of power and express the evolution along P P V P j=0,...,M i,j m P V P = L ; (SY2-Bis) the periods of the load i V of the PVP battery. i P ∗ – ∀ j = 0 , . . . , M : p · i · U = τ ; (SY3) • i=0,...,N −1 i,j j The EV Constraints : They contain VR1, ..., VR6-Bis, together with constraints related to the time and power P V P EV m – ∀i = 0, . . . , N − 1, j = 0, . . . , M: i,j i,j ≤ Inf ( C , C ) · U ; (SY4) values when the vehicle visit the customers or to PVP. • The Recharge Event Constraints: They link together the PVP periods and the time horizon [0, p · N ] of EV. Theorem 2: Above VR EPC MILP MILP model solves Those constraints come as follows: VR EPC in an exact way. VR EPC Constraints and Objective Function: P We handle VR EPC MILP through Branch and Cut. • Objective: Minimize (Ψ i.δi + Φi(yi) + α.τM+1 i • PVP Constraints C. Numerical Experiments – ∀i = 0, . . . , N : V ≤ C ; (PC2) i Technical Context: The experiments are performed on an P V P P V P P V P P V P – V 0 = H 0 ; V ≥ H ; (PC3) N 0 AMD EPYC 7H12 64-Core processor, under Gnu/linux P V P – Ch ∀ i = 1 , . . . , N − 1 : y i i Purpose ≤ C · (1 − δ ); (PC1) : Evaluating the P V P VR EPC MILP MILP model. P V P – P V P ∀ i = 1 , . . . , N : V = V + i i−1 Ubuntu 20.04.2CPLEX 12.10 is used in single-thread mode. y P V P i i Instances: − L ; (PC4) The main parameters of every instance are the customer number M (from 5 to 30), the period number N • EV Constraints (from 40 to 320), and the period length p ∈ {1, 2, 4}. We – (VR1, ..., VR6-Bis) involved in Lemma 1; derive instances with p = 2 (Id − 2) from instances with – EV EV EV EV V = H ; V ≥ H (VR7) p = 1 (Id) by merging the periods, p · N remaining the same. ; 0 0 M +1 0 – EV EV ∀ j = 0 , . . . , M + 1 : E j, −1 j ≤ V ≤ C ; (VR8) – ∀j, k = 0, . . . , M + 1: X j,k → Results: For every instance, Table II displays the lower bound ( EV V +E +(Z (E k j,k j,k j, − − 1) ·1 −1,k j,k and the upper bound + E − E )) ≤ LBG LP U BG U B computed through ( EV EV V + L ) ; (VR9) Branch and Cut by CPLEX library in 2 CPU hours. j j 58 IV. HEURISTIC HANDLING OF VR EPC TABLE II RESULTS FOR GRASP VR EPC We drive the leader route Γ (the vector X of the MILP setting) while relying on the correlation which exists between the Id GRASP 150 Desc + LS T Desc T GRASP optimal value of VR EPC 3 173 165 122,7 145.6 ( Γ ) and some key features of Γ . Let 3-2 176 171 35,6 47.6 T E us W (Γ) be the optimal value VR EPC ( Γ ) and L (Γ) , L (Γ) 5 156 168 356,2 327.2 be the lengths of Γ in the sense of T and E respectively. 6 175 176 565,7 845.1 Standard TSP 5-2 158 173 109,8 134.1 2 Opt and Reloc operators act on any route Γ through 2 parameters 6-2 181 181 203,6 227.2 j , j in J + { Depot } : 1 2 8 121 121 1998,6 542.5 • 2 Opt (Γ, j1, j2) replaces the moves from j1 (j2) to its 9 119 122 2351,0 578.9 • successor ¯ ¯ ¯ ¯ 8-1 132 134 773,0 246.7 j 1 ( j 2 ) by moves from j 1 ( j 1 ) to j 2 ( j 2 ); 9-1 123 130 629,1 189.3 ¯ Reloc (Γ , j 1 , j 2 ) relocates j 1 between j 2 and j 2 . A. Approximating W (Γ) through Path Search T The Pseudo-Descent Algorithm Descent VR EPC: We al- low E 2 Opt and Reloc to slightly deteriorate L (Γ) and L(Γ). Once Γ is fixed, a full solution is determined by the sequence T E Given 2 parameters δ > 0 , δ > 0 and 2 routes Γ1, Γ2, of recharge events linking EV and PVP, augmented with the T E we say that Γ 2 deteriorates Γ 1 by no more than ( δ , δ) if amounts of power bought between 2 consecutive recharge T T E E L (Γ 2 ) − L (Γ 1 ) ≤ δ 1 or L (Γ 2 ) − L(Γ1) ≤ δ2. Then the transactions. Thus, we design a PDYN EPC algorithm that T E Descent VR EPC ( δ , δ) algorithm works as follows: searches for a path in a transition graph: Γ 1) Initialize by applying GRASP SVR EP(1); 2) At any iteration do • A state in the sense is a 4-uple V EV P V P S = ( j, i, V , V ) j i a) Generate all the , EV 2 Opt and Reloc parameters where j is a customer, i a period, V is the power load j (j1, j2) such that resulting route deteriorates Γ by P V P of EV when it leaves j and V i the power load of PVP T E no more than ( δ , δ ) ; at the end of i, with the implicit meaning that a recharge b) For any (j1, j2) selected this way and any re- transaction involving i and j has just been performed. sulting route Γ 1 compute W Aux(Γ1) through • A decision is a 4-uple (j 1, i1, m1, y ¯), where (j1, i1, m1) PDYN EPC; If improving Γ is possible, then do means the next recharge event and y ¯ means the power it according to a Best Descent strategy else stop. bought by PVP during the periods i + 1, . . . , i1 − 1. Related transition cost is C. Numerical Experiments p · ( i 1 − i ) augmented with the purchase cost of y ¯, which corresponds to the opti- Table III involves the same instances as Table II. It displays: mal value of some convex optimization program and is • The value GRASP 150 obtained by computed and stored as part of a pre-process. GRASP VR EPC(150) and related CPU time • EV P V P Initial state S 0 0 is a 4-uple ( Depot, − 1 , H , H ). T GRASP . 0 A final state is any 4-uple EV ( Depot, N − 1 , V ≥ j • The value Desc+LS by pipelining: GRASP VR EPC(1) H EV P V P P V P , V ≥ H ). (1 replication) → Descent VR EPC(4, 4), together with 0 j 0 CPU time T Desc. We speed the algorithm by fixing an upper bound N Dec on the Comments: GRASP VR EPC with 150 replications number of feasible decisions and introduce filtering devices. most often reaches quasi-optimality. The pipe-line GRASP VR EPC(1) → Descent VR EPC(4, 4) improves B. Two Simple Heuristic Algorithms resulting value in 2 among the 10 instances. Experiments show that the evolutions of T W (Γ) , L (Γ), L E(Γ) and the number N RT during a decent loop performed REFERENCES on W (Γ) via the 2 Opt and Reloc operators are strongly correlated. So we partially short-cut T [1] Drexl M.: Synchronization in vehicle routing–a survey; Transportation W (Γ) and rely on L (Γ) and E Science 46, pp. 297-316, (2012) L (Γ) in order to drive Γ towards good solutions. [2] Erdelic T., Caric T., Lalla-Ruiz E.: A Survey on the Electric Vehicle Routing Problem.Journal of Advanced Transportation (2019). The GRASP Algorithm [3] Koc¸ C., Jabali O., Mendoza J., Laporte G.. The electric vehicle routing GRASP VR EPC :It considers a replication parameter problem with shared charging stations. ITOR, 26, (2018). Q and for any q = 1 , . . . , Q : [4] Kuo Y.. Simulated annealing to minimize fuel consumption for the time- 1) It randomly generates parameters T E Comp. Indust. Eng. ω , ω ≥ 0 together dependent VRP., 59 :157–165, (2010). with an initial [5] Lajunen A.: Energy consumption and cost analysis of electric city buses. Γ( q ) ; Transportation Research Part C , 38 :1–15, (2014). 2) It applies 2 Opt and Reloc until Γ(q) becomes a local [6] Luthander R., Widen J., Nilsson D., Palm J.: Photovoltaic self- optimum with respect to T T E E consumption in buildings. Applied Energy 142, p 80-94, (2015). ω · L (Γ)+ ω · L (Γ) ; 3) It computes [7] Macrina G., Di Puglia L., Guerriero F.: The Green-Vehicle Routing W Aux (Γ( q )) via PDYN EPC and updates Problem: A Survey. Model. and Optim. in Logistics p 1-26 (2020). the best current route Γ(qBest). [8] Verma A.: Electric vehicle routing with recharging stations. Euro Journal of Transport. Logistics 7, p 415-451, (2018). 59 A discrete event simulation model for analyzing the wood waste reverse supply chain Nikola Kovačević Črtomir Tavzes Balázs Dávid nikola.kovacevic@f amnit.upr.si crtomir.tavzes@innorenew.eu balazs.david@innorenew.eu UP FAMNIT, University of InnoRenew CoE, UP IAM, InnoRenew CoE, UP IAM and UP Primorska University of Primorska FAMNIT, University of Primorska Koper, Slovenia Koper, Slovenia Koper, Slovenia Abstract Reverse supply chains for wood waste operate within the framework of cascading use principles, where reclaimed wood un- This paper presents a discrete event simulation model for analyz- dergoes successive down-cycling through high-value applications— ing reverse supply chain operations of wood waste materials. The from structural reuse to particleboard feedstock—before final model enables comprehensive evaluation of storage management, energy recovery, thereby maximizing carbon storage while re- processing capabilities and resource allocation in reverse logis- ducing virgin material demand [3]. tics networks, tracking the flow and transformation of resources Wood waste reverse supply chains involve diverse stakehold- from generation to its final destination either as recovered and ers spanning sawmills, panel producers, demolition firms, recy- reused products of landfilled waste. The proposed model is capa- clers, transport providers, and municipal waste collectors. These ble of managing different inventory policies and stock strategies. actors coordinate complex logistical processes including collec- The efficiency of the proposed model is shown on a network tion, transportation, quality control, sorting, and processing op- generated based on statistical data of waste wood in Slovenia. erations. However, this complexity introduces substantial chal- Keywords lenges: seasonal and spatial dispersion of biomass sources, wide variability in material form and contamination levels, high pro- reverse supply chain, discrete event simulation, inventory man- cessing costs, and insufficient information systems for post-consumer agement, wood waste wood recovery [7]. Discrete Event Simulation (DES) has emerged as a particularly effective tool for modeling these complex reverse logistics sys- 1 Introduction tems. Unlike analytical methods that often require simplifying Wood waste represents a growing environmental and economic assumptions, DES can capture the stochastic nature of return challenge worldwide, driven by expanding construction, man- flows, processing variability, and resource constraints that char- ufacturing, and demolition activities. If not managed properly, acterize real-world reverse logistics operations [4]. it constitutes both a lost resource and a significant burden, con- Discrete-event simulation has proven effective for studying tributing to greenhouse gas emissions, pollution, and rising land- supply chain dynamics and transport logistics, particularly in fill costs. Recent years have witnessed increased wood waste gen- complex systems where uncertainty and stochastic elements play eration due to expanding construction sectors, including substan- critical roles [8]. By representing a supply chain as a sequence tial increases in renovations and refurbishments for structural of discrete events, DES enables detailed representation of opera- and energy improvements of building stock, alongside growing tional policies and aids in evaluating alternative strategies and demand for wood-based packaging [14]. system responses to uncertainty. Recent advances demonstrate The environmental and economic consequences of wood waste DES capability to analyze complex multi-level resilience relation- mismanagement are profound. Conventional disposal methods, ships that traditional analytical methods struggle to capture [6]. such as landfilling and incineration, release harmful greenhouse Current research predominantly employs deterministic opti- gases like methane and carbon dioxide, while slow decomposition mization models that fail to capture the inherent stochasticity under anaerobic conditions creates long-term environmental haz- of wood waste systems. While existing research has advanced ards [11, 10]. Beyond ecological damage, improper disposal poses understanding of individual system components, there remains public health risks due to associated pollution. Conversely, wood a gap in DES applications to wood waste reverse supply chains. waste remains a vastly underutilized resource, with substantial The bidirectional flows, uncertain material quality, and variable volumes that could be repurposed into valuable raw materials or availability patterns characteristic of these systems require simu- energy sources if managed effectively [2, 9]. lation approaches capable of representing stochastic interactions To fully realize these opportunities, effective reverse supply and multi-stakeholder coordination mechanisms. chains are essential. Unlike traditional forward supply chains, This paper presents a DES model designed for analyzing re- which move products from raw materials to consumers, reverse verse supply chain operations in wood waste management con- supply chains focus on recovering value from used products texts. The model enables evaluation of processing capabilities, through collection, reprocessing, and redistribution [5, 12]. Wood resource allocation strategies, and operational policies through waste presents unique logistical and environmental challenges, a dual-axis inventory management framework that combines including uncertain return volumes, variable product conditions, push/pull strategies with different stock management approaches. and complex processing requirements, but also significant poten- Using Slovenia as a case study, we demonstrate how the simu- tial for energy recovery and impact reduction. lation framework supports evidence-based decision-making in reverse logistics network design and management. MATCOS 2025, 9–10 October 2025, Koper, Slovenia 2024. 60 MATCOS 2025, 9–10 October 2025, Koper, Slovenia Kovačević et al. Product A Demand needs waste P,Q,R order Treatment Inventory Treatment Handle excess product A transport (convert waste (buy inventory or landfill) into product A) excess manage waste type P inventory storage order transport Collector transport (waste types excess P,Q,R) manage transport inventory storage excess order transport generate Generator generate Generator generate Generator (waste type R) (waste type Q) (waste type P) manage manage manage inventory storage storage storage inventory inventory Figure 1: Overview of the model. 2 Model Development flows. Collected material is stored in finite-capacity collection centers that decouple generation events from treatment schedul- A discrete-event simulation model was developed for the repre- ing. The transport system operates at two levels: local collection sentation of the wood waste reverse supply chain. The proposed from generators to collection centers with capacity and distance framework models waste flows from generation through col- constraints, and long-haul transport from collection centers to lection to final processing across connected regions, with each treatment facilities using priority-based scheduling. region containing generators, collectors, and treatment facilities. The system tracks three primary entity categories that interact Treatment Operators represent industrial facilities that serve as final waste customers and value-adding stages. These enti- through event-driven communication protocols, responding to ties consume waste as raw material, performing transformation state changes and resource availability. according to predefined recipes for products. Figure 1 provides an overview of this system, presenting all model entities, their connections and the possible decisions avail- able to them. Table 1: Policy Configuration Framework 2.2 Decision strategies Configuration Description All entities operate under a dual-axis operational paradigm that collectively dictates production and procurement logic through- P USH ON DEMAND Continuous forecast-driven production out the reverse supply chain. The axis defines Inventory Policy P USH REORDER 50 Production triggered at 50% capacity strategic philosophy: P USH policies operate using forecast-driven P USH REORDER 90 Production triggered at 90% capacity approaches, managing operations based on internal state projec- P ULL ON DEMAND Demand-triggered operation only tions, while P ULL policies implement lean, demand-driven ap- P ULL REORDER 50 Demand-driven with 50% threshold Stock Strat- proaches aligned with Just-in-Time principles. The P ULL REORDER 90 Demand-driven with 90% threshold egy axis defines tactical execution rules: ON DEMAND repre- sents continuous operation under P USH or operation only upon signal receipt under P ULL, while buffer-based strategies (RE- 2.1 Entity Modeling ORDER 50 and REORDER 90) trigger actions when inventory Waste Generators represent real-world actors that produce levels cross 50% or 90% capacity thresholds respectively. This wood waste, serving as material sources introducing specified framework generates six distinct operational models (2×3) that quantities and types over time. Generator behavior incorporates enable systematic empirical comparison. variability through random seasonal fluctuations reflecting mar- Table 1 summarizes the six operational configurations evalu- ket and operational uncertainties. Generators produce waste ated in this study. Each configuration represents a unique combi- proactively regardless of system policy. nation of inventory policy (P USH vs P ULL) and stock strategy Collector Companies serve as commercial intermediaries (ON DEMAND, REORDER 50, REORDER 90), enabling analysis operating as midstream buffers managing bidirectional material of different supply chain management approaches. 61 A discrete event simulation model for analyzing the wood waste reverse supply chain MATCOS 2025, 9–10 October 2025, Koper, Slovenia The framework incorporates realistic capacity constraints with OSB demonstrating the highest carbon storage potential through an overflow management system. When entities ex- ( 1213.60 kgCO2e/m ). − 3 ceed storage capacity, they choose between capacity expansion While not all waste codes contribute to product creation, un- or landfill disposal using cost-minimization logic with dynamic used codes were retained to maintain representation of Slovenia’s pricing that escalates with repeated use, reflecting real-world wood-related waste streams, and resources of these waste codes constraints and discouraging reactive management. still contributed to decisions about inventory management and landfilling. Initial generation rates derived from empirical data by 2.3 Stochastic Elements the Slovenian Environment Agency (ARSO) datasets [1] and un- derwent dynamic adjustment throughout the 365-unit simulation The model incorporates several stochastic components to cap- period. ture real-world uncertainty. Entity failures use uniform recovery durations with failure probabilities varying by scenario severity. Waste generation includes daily variability factors from clipped 3.2 Inventory Management Performance normal distributions, while treatment conversion efficiency is Storage utilization analysis reveals distinct patterns across entity normally distributed around base values. The model ensures re- types and strategies. P USH strategies maintain consistently high producibility through hierarchical random number generation storage utilization with frequent saturation events, particularly with deterministic seeding for consistent behavior across runs. in the simulation’s latter half, reflecting proactive inventory pre- positioning regardless of downstream demand. P ULL strategies 3 Results demonstrate more dynamic, demand-responsive behavior with The simulation study analyzes six operational models across a lower average utilization and greater temporal variability. 365-day simulation period representing one operational year. At the generation level, P USH policies show progressive stor- Each configuration underwent 100 Monte Carlo replications with age accumulation leading to widespread saturation across most unique random seeds to capture stochastic variability. regions, with particularly high utilization (80-100%) sustained System performance is evaluated through five complementary throughout the simulation period. P ULL strategies maintain more metrics that capture operational effectiveness across the complete balanced levels with moderate utilization (20-60%) and better material flow trajectory. demand synchronization, avoiding the excessive accumulation Service level quantifies system reliability by measuring the patterns observed in P USH configurations. percentage of generated waste successfully converted into final Collection storage exhibits similar patterns, with P USH poli- products, representing the fundamental waste diversion capa- cies creating sustained high utilization across all regions and bility. Operational efficiency provides hierarchical assessment stock strategies, while P ULL policies show significantly lower through three interconnected ratios: collection efficiency (waste average utilization with more efficient turnover. The temporal collected relative to generated), processing efficiency (waste con- analysis reveals P USH strategies experience early saturation that verted relative to collected), and overall system efficiency (final persists throughout the simulation, whereas P ULL strategies products relative to initial generation), thereby identifying bot- maintain responsive storage levels that adapt to demand fluctua- tlenecks and resource utilization patterns throughout the sup- tions. ply chain. Storage utilization examines capacity management by tracking current storage usage against maximum capacity 3.3 Cost-Environmental Impact Analysis across collection centers and processing facilities, revealing both The cost-environmental impact analysis, based on single simu- potential constraints and the effectiveness of demand-supply syn- lation runs for each configuration, reveals significant trade-offs chronization under different inventory policies. Environmental across supply chain configurations. P ULL REORDER 90 emerges impact quantifies the emissions in kgCO e equivalent generated 2 as the best-performing strategy for environmental performance, through transportation, processing operations, and landfill dis- achieving the lowest environmental impact (approximately posal activities. Finally, landfill overflow measures the volume of 800k kgCO2e) at competitive costs (around 30 million €). P ULL waste diverted to disposal due to system capacity limitations or ON DEMAND demonstrates comparable performance with slightly processing constraints, representing both environmental burden higher emissions (approximately 970k kgCO 2e) at lower cost lev- and foregone resource recovery opportunities. els. P ULL REORDER 50 occupies an intermediate position within Collectively, these metrics provide comprehensive assessment the P ULL solution space, demonstrating suboptimal efficiency of reverse supply chain performance across operational, environ- relative to alternative configurations due to higher costs despite mental, and resource recovery dimensions. moderate environmental performance. The simulation framework was implemented with SimPy as P USH strategies generally show higher environmental im- the core simulation engine [13]. pacts, with P USH REORDER 90 representing the least favorable 3.1 option, generating approximately 2m kgCO2e emissions at costs Setup around 58 million €. The analytical framework employed Slovenia as the case study, Figure 2 illustrates the total costs and environmental impacts with each of the 12 statistical regions of the country represented of each strategy, demonstrating P ULL policy as the dominant by a single instance of each system entity (generator, collector policy across both cost and environmental dimensions. and treatment). The system incorporated 14 waste codes and 3 product types. These types were selected to be engineered wood products: 3.4 Monte Carlo Analysis Results Particle Board (accepting five waste types), OSB (four waste The statistical analysis based on 100 Monte Carlo replications types), and MDF (four waste types). Production prioritization reveals different performance patterns when accounting for sto- follows ABC analysis based on total biogenic carbon impact, chastic variability. All strategies achieved high service levels 62 MATCOS 2025, 9–10 October 2025, Koper, Slovenia Kovačević et al. achieving better demand synchronization and reduced waste push - on demand overflow compared to forecast-driven P USH approaches. 2M These findings suggest that waste management practition- reorder 90 push - reorder 90 push - reorder 50 pull - on demand ers should prioritize demand-driven operational models over pull - reorder 90 1.8M pull - reorder 50 forecast-based strategies. P ULL REORDER configurations emerge as particularly effective, balancing high service levels with mini- 1.6M mal environmental impact through responsive inventory manage- 1.4M on demand ment that avoids excessive accumulation and associated disposal costs. reorder 50 vironmental Impact (kg CO₂e) 1.2M The framework provides valuable insights for reverse logistics network design and demonstrates the importance of aligning Total En on demand 1M inventory policies with sustainability objectives in waste man- reorder 50 rder 90 0.8M agement systems. Future work should extend the model to incor- 30M 35M 40M 45M 50M 55M 60M porate multi-product interactions, dynamic pricing mechanisms, Total Cost (€) and regional policy variations to enhance applicability across diverse waste management contexts. Figure 2: Cost versus environmental impact analysis across supply chain configurations. Acknowledgements The research was supported by the BioLOG project: the authors are grateful for the support of National Center of Science (NCN) through grant DEC-2020/39/I/HS4/03533, the Slovenian Research above 97%, with P ULL REORDER 50 reaching the highest av- and Innovation Agency (ARIS) through grant N1-0223 and the erage (99.9%), followed closely by P ULL REORDER 90 (99.7%). Austrian Science Fund (FWF) through grant I 5443-N. Balázs P USH strategies showed slightly lower but still strong service Dávid is also grateful for the support of the Slovenian Research levels, ranging from 97.8% (P USH ON DEMAND) to 98.6% (P USH and Innovation Agency (ARIS) through grant J1-50000. Balázs REORDER 90). Dávid and Črtomir Tavzes gratefully acknowledge the Slovenian Environmental impact analysis reveals P ULL REORDER 50 Research and Innovation Agency (ARIS) and the Ministry of the as the most environmentally friendly option (278k kgCO2e), fol- Economy, Tourism and Sport (MGTŠ) for the grant V4-2512. lowed by P ULL REORDER 90 (323k kgCO2e). P USH REORDER 90 generated the highest emissions (629k kgCO 2e), more than References double that of the best-performing configuration. [1] Agencija Republike Slovenije za okolje (ARSO). 2025. Poročila in publikacije Landfill overflow analysis demonstrates superior waste man- o odpadkih. (2025). Retrieved Aug. 20, 2025 from https://www.arso.gov.si/v agement performance of P ULL strategies. P ULL REORDER 50 arstvo%20okolja/odpadki/poro%C4%8Dila%20in%20publikacije/. [2] Peter Akhator, Albert Obanor, and Anthony Ugege. 2017. Nigerian Wood achieved zero landfill overflow, while P ULL ON DEMAND gen- Waste: A Potential Resource for Economic Development. Journal of Applied 3 erated minimal overflow (16.1 m ). In contrast, P USH REORDER Sciences and Environmental Management , 21, 2, 246. doi: 10.4314/jasem.v21i 3 90 produced substantial landfill overflow (2,014 m ), indicating 2.4. [3] Michael Burnard, Črtomir Tavzes, Aleksandar Tošić, Andrej Brodnik, and significant waste management inefficiencies. Andreja Kutnar. 2015. The role of reverse logistics in recycling of wood Efficiency comparison reveals nuanced performance patterns products. In Environmental Implications of Recycling and Recycled Products. across configurations. Collection efficiency ranges from 39.3% Subramanian Senthilkannan Muthu, editor. Springer Singapore, 1–30. doi: 10.1007/978- 981- 287- 643- 0_1. (P ULL REORDER 50) to 58.1% (P USH REORDER 90), while pro- Discrete-Event Simulation [4] George S. Fishman. 2001. . Springer New York, cessing efficiency shows P ULL strategies generally outperform- New York, NY. isbn: 978-1-4419-2892-4 978-1-4757-3552-9. doi: 10.1007/978 - 1- 4757- 3552- 9. ing P USH approaches, with P ULL REORDER 50 achieving the [5] S.M. Gupta, ed. 2013. . (1st ed.). Reverse Supply Chains: Issues and Analysis highest processing efficiency (38.2%). Overall system efficiency CRC Press. doi: 10.1201/b13749. demonstrates P USH REORDER 90 as the top performer (19.0%), [6] Dmitry Ivanov. 2025. Comparative analysis of product and network sup- ply chain resilience. , International Transactions in Operational Research followed by P ULL REORDER 90 (16.6%) and P ULL REORDER 50 itor.13612. doi: 10.1111/itor.13612. (14.9%). [7] Arkadiusz Kawa. 2023. REVERSE SUPPLY CHAIN OF RESIDUAL WOOD BIOMASS. . Logforum [8] C. Kögler and P. Rauch. 2018. Discrete event simulation of multimodal and 4 Conclusion & Future Work unimodal transportation in the wood supply chain: a literature review. Silva Fennica, 52. doi: 10.14214/sf .9984. In this paper, we developed a DES framework for analyzing waste [9] Dorin Maier. 2023. A Review of the Environmental Benefits of Using Wood and material flow and transformation in reverse supply chains. Waste and Magnesium Oxychloride Cement as a Composite Building Mate- rial. , 16, 5, 1944. doi: 10.3390/ma16051944. Materials The proposed model was tested on a simplified network of wood [10] J. Owoyemi, H. Zakariya, and I. Elegbede. 2016. Sustainable wood waste waste flows built on statistical data by ARSO. The model is ca- Environmental & Socio-Economic Studies management in nigeria. , 4, 3, 1–9. pable of evaluating different operational strategies through a doi: 10.1515/environ- 2016- 0012. [11] Kemal Parlak, Nural Yilgor, and Atakan Öngen. 2024. Hydrogen-rich syngas dual-axis framework that combines inventory policies (P USH production from wood waste and wood waste pellet via gasification in vs P ULL) with stock management approaches (ON DEMAND, updraft circulating fixed bed reactor. Research Square, (May 2024). Preprint. REORDER 50, REORDER 90), enabling systematic comparison of doi: 10.21203/rs.3.rs- 4343729/v1. [12] Carol Prahinski and Canan Kocabasoglu. 2006. Empirical research opportu- supply chain configurations under realistic stochastic conditions. Omega nities in reverse supply chains. , 34, 6, 519–532. doi: 10.1016/j.omega The simulation results demonstrate that P ULL-based inven- .2005.01.003. [13] Team SimPy. 2025. Simpy. Accessed: 2025-05-15. (2025). https://simpy.readt tory strategies offer superior environmental performance while hedocs.io/en/latest/. maintaining competitive service levels. The analysis reveals a [14] Statistical Office of the Republic of Slovenia. 2024. Več komunalnih odpadkov fundamental trade-off between aggressive inventory accumu- predvsem zaradi povečanja količine kosovnih odpadkov. Stat.si. Accessed: 2025-05-15. (2024). https://www.stat.si/StatWeb/News/Index/12770. lation and environmental sustainability, with P ULL strategies 63 Robust (re)Design of Material Flow in Circular Networks – a Scientific Approach Ádám Szaller* Balázs Dávid Péter Egri Miklós Krész József Váncza InnoRenew CoE, InnoRenew CoE, HUN-REN Institute HUN-REN Institute HUN-REN Institute UP IAM and UP UP IAM and UP for Computer for Computer for Computer FAMNIT, University FAMNIT, University Science and Control Science and Control Science and Control of Primorska of Primorska Budapest, Hungary Koper, Slovenia Budapest, Hungary Koper, Slovenia Budapest, Hungary balazs.david@ miklos.kresz@ adamszaller@sztaki.hu egri@sztaki.hu vancza@sztaki.hu innorenew.eu innorenew.eu Abstract levels of circularity [7][28]. Reverse logistics methods can be integrated into the conventional process of supply chains, As sustainability concerns grow, the conventional linear supply forming a circular or closed-loop supply chains that account for chain model — where products move in a single direction toward both forward and reverse flows of resources [19]. As consumers — is proving insufficient. In Circular Networks sustainability is becoming a key issue, the main arguments for a (CNs), forward and reverse flows of resources coexist and should transition towards circular value chains are linked to the be jointly optimized. Yet, CNs face significant challenges, increasing scarcity of natural resources and the ecological impact including uncertainty in the availability, quality, and usability of of human activities. Such a sustainable mindset has become end-of-life products, as well as unpredictable demand for crucial for the supply chains of several significant materials, such recovered materials. These factors increase risks in processing as steel and aluminium, plastics [23], bottles [12], textiles, paper and limit adaptability, since traditional supply chains are rigid or wood. Despite the increasing need of circularity, there are still and rely on stable partnerships. While resilience strategies and numerous research and practical gaps in literature [27]. However, optimization methods such as stochastic or robust models have there are several issues that make the optimal design of material been explored, they often oversimplify real-world dynamics and flows in such Circular Networks (CNs) exceptionally neglect circular flows. To address these gaps, this paper challenging. Besides the hard-to-predict demand for products introduces a block-based value chain model and a 5-step and components, the potential reuse of end-of-life products and scientific approach. production residues introduces new sources of uncertainty on the Keywords supply side. Not only is the available quantity for end-of-life products unknown in advance, but their quality and usability are robust material flow planning, circular networks also questionable, introducing yield uncertainty to the waste processing. The same is true for production residues; while types 1 and quantities can be estimated for each production process, their Introduction ultimate quality will depend on the original input materials to Flow of products and provision of services between suppliers and their corresponding processes. The recovery of the used products consumers is an integral part of modern-day business processes. is also a challenging process with difficult, complex and This has traditionally been the domain of supply chain expensive steps. For example, recovered wooden products could management, in which different actors work together to improve require sorting, removing of tacks, nails, paint and even bugs. operational efficiency by sharing information and linking their There might be possible different ways to reuse these products activities. As environmental awareness became an increasingly which degrade the material quality to a different extent, such as important topic, an alternative was needed to the traditional using as solid wood, particle board, fiber-based product, forward product flows that only point towards the consumer. chemical product or even incineration as an energy source [15]. According to Dekker et al. [8], more emphasis had to be put on Moreover, supply chains are traditionally linear and static, and end-of-life recovery and the reuse of resources in the supply their entities usually cooperate with their well-established chain due to different driving forces (economic, legislative, and connections, always relying on the same pool of actors and corporate citizenship). The field of reverse logistics is the processes [29]. Because of this, their adaptation to rapid changes integration of all activities that aim to recover resources from is inefficient, and disruptions either on supply- or demand-side, their final destination either to produce further value or to dispose or in the processes of the system itself can cause issues. of them properly. This can be done either in a direct manner (resale/reuse/redistribution) or through a more complex recovery process (repair, reuse, refurbish, remanufacture, retrieve, recycle, incinerate/landfill), which are described in the 10 R model of the 64 2 Research objectives  Develop a robust design method for material flow in circular supply chains. Research of dynamic and resilient supply chains has started in  Analyze the impact of disruptions (on demand, supply, or recent years [16], and it has been identified that dynamics control process side) on these robust flows. can be classified into proactive (designing in a redundant and  Create efficient methods for disruption management and flexible way) and reactive (recovery methods in case of material flow redesign in the network. disruptions) stages. While there are several different  Ensure general applicability of the methods developed categorizations of supply chain disruptions, the ones that appear across various circular industrial use-cases. consistently in every group are process risks, demand risks and supply risks [18]. Such uncertainties in circular supply chain 3 Scientific approach planning are usually tackled with stochastic and robust optimization approaches, such as chance constraints, scenario- 3.1 Block-based model representation based optimization, data-driven robust optimization and maximin optimization [2][20][25][30]. In addition to the various First, this paper proposes a modular modeling concept that can sources of uncertainty, there are multiple, often contradicting be applied to represent any supply chain in general, incorporating objectives of the supply chain planning problem. Supply chain both forward and reverse material flows. With the help of this planning traditionally aims at satisfying the demand at minimal concept, a multi-commodity flow network can be constructed cost, but this does not necessarily meet other objectives, such as using the existing entities of multiple independent value chains, an increased proportion of recycled materials or a minimal value and material flow and transformation can be optimized in this loss during the recovery process (the so-called biomass network. This representation is also capable of accommodating cascading principle [15]). Furthermore, conflicting objectives cascade or reuse possibilities and residual material flows. The can also arise from the presence of multiple stakeholders in the value chains are deconstructed into independent entities called circular chain [13]. blocks. A block is a universal representation of a series of Due to these complexity and uncertainty issues, the performance different activities inside a value chain, defining the required of the circular business models must be extensively explored material inflow and outflow, as well as the material before implementation, i.e., with simulation transformation happening during the corresponding processes. models [21]. Despite the growing need for models and tools for assessment of Material flows are described by a vector of features, and circularity, simulation approaches for circular supply chains are outflow(s) can be directed to be inputs of another block if the still scarce in the literature and in practice. According to a recent features of the flow matches the input requirements of the block. systematic review, most models apply optimization or multi This feature vector can include all relevant material criteria decision making models which greatly outnumber characteristics (depending on the application), as well as simulation approaches [22]. economic, social and environmental costs and parameters. A In our previous joint work, we have focused on robust design of general outline of this modeling approach is given in Figure 1. waste wood collection and transportation networks [10], which we intend to continue with the modeling, planning and validating robust supply chains considering multiple optimization criteria. We have also developed a robust two-stage model for planning and adapting the supply chain material flow to the disturbances [11]. On the level of mathematical abstraction, supply chains can be regarded as multi-commodity flow networks where the flow (resources) is transformed into the nodes of the system. While the theory of robust and adaptive network flows has been studied in the past [4][5], and they have been proven to be NP-hard problems [9], the application of this theory to actual supply chain networks has been limited. Even recent studies [1][14] simplify the real-world aspects and dynamics of supply chains significantly to a level where their results—while being important theoretical contributions—cannot be used directly for real-world networks. Moreover, none of these studies consider reverse or circular flows to our knowledge, they only deal with the traditional forward case. Based on the above-mentioned research challenges, the Figure 1: Graphical representation of the block-based objectives to be solved with the scientific approach presented in model elements this paper are the following:  Address shortcomings of existing approaches for circular The modeling concept distinguishes between three different supply chains. block types: resource, process, and product/demand blocks. This  Design a novel, general approach for modeling material corresponds to most value chains, where initial resources are flow in reverse and circular networks. transformed by a sequence of processes and yield products that 65 satisfy some customer demand. Resource blocks (Figure 1.b) and connected to the yield uncertainty with regards to residual provide data about the initially available resources in the network materials of the processes in the network. and can represent both virgin and reused materials. Process blocks Following the design of the material flow, a novel method must (Figure 1.c) can represent any activity inside the chain transformation of the material properties during the activities unforeseen events to redesign the flow to still satisfy the original constraints as fully as possible. These unforeseen events can they represent, as well as any additional outputs (residuals, etc.) (production, transportation, storage, etc.) and also define the be developed for disruption management and recovery after they might have. The chain ends with demand blocks (Figure materials), demand, or in the production processes of the system, include changes in the original supply (both for virgin and reused 1.d), which represent the needs in material quantity/quality of the leading to a lower yield of reverse materials to be reused. The end-users (customers, factories, etc.). With this modeling three types of disruptions must be examined independently, as approach, existing value chains (Figure 1.a) can be de-linearized well as simultaneously, studying their combined effect on the and connected into a common supply chain networks by system. Literature for optimization methods in reactive combining all the possible directions of resource flow (both disruption management (re-planning after an unforeseen event) above modeling approach, a supply network can be constructed traditional forward chains [17]. Metaheuristic approaches such as Tabu search or Simulated Annealing, as well as Large using the entities (processes) of multiple existing value chains as primary and residual) and transformation processes. Using the for supply chains is quite limited and instance-based even for Neighborhood Search are promising approaches in this topic. nodes, where all possible material flow options between these actors can be represented. Figure 2 shows an example for four these algorithms and new diversification methods can be Various new neighborhood structures can be constructed for different linear value chains (I-IV), with the potential reverse implemented for exploring new regions of the neighborhood. flow options of residual materials from the processes of one The hybridization possibilities of metaheuristic and exact value chain used as inflow in the processes of other chains. approaches must also be studied, such as aiding the branch and bound solution of the mathematical models with a rounding heuristic or solving the subproblems of a large neighborhood search by exact methods. The applicability of Genetic Algorithms and Ant Colony Optimization must also be investigated to find a suitable and efficient solution. To validate and further develop the above-mentioned methodologies, a real-world use case is proposed using e.g. the wood industry as a basis. Wood is an ideal example for a circular material, as both production residues and recovered end-of-life products can be reused as inputs for other lower value wood products. Actors and processes from the value chains of multiple wooden products (e.g. furniture, panels, packaging) could be Figure 2: Circular resource flow in the supply network modeled using the block-based network, and the developed models and methods could be validated on this use-case. 3.2 Research steps As a final step, simulation methods could be applied to further evaluate the efficiency of the disruption management methods on Based on the above approach, the following research steps are large amount of realistic, generated data (for which the related proposed. literature and experience from industry can give a basis). First, a novel and universally applicable mixed-integer linear 4 Conclusion and future work programming model must be formulated for circular material flow optimization using the flexible block-based network In the paper, a scientific approach regarding robust (re)design of representation. The major challenge of this step is incorporating material flow in circular networks was introduced. First, a block- flow transformation functions within the (process) nodes of the based representation – that takes into account several aspects of network, corresponding to the production processes of the given forward and reverse material flows by defining resource, process, entities. These possible production choices for flow and product/demand blocks, and the vector of features connected transformation must be reflected in the decision variables to each block – was presented to model the circular value chain. managing material flow between the nodes of the network, as Then, a 5-step scientific approach was suggested for the well as their connected constraints. The model must consider (re)design process: 1) creating mixed-integer linear multiple objectives (e.g. operational costs and environmental programming model, 2) designing and optimizing robust impacts). material flow, 3) developing a methodology for disruption management and recovery, 4) validating on a real-world use case, Next, a method for optimal design of robust material flow 5) creating a simulation model and testing with generated data. optimization must be created for this network, e.g. by adapting a selected method from literature. The extension of the robust Acknowledgements modelling approach proposed by Bertsimas and Sim [6], and the two-stage robust network flow approaches such as the one by This research has been supported by the TKP2021-NKTA-01 Atamturk and Zhang [3] and especially by Simchi-Levi et al. [26] NRDIO grant on "Research on cooperative production and are considered as promising starting points. The developed logistics systems to support a competitive and sustainable method must take into account possible uncertainties on the economy". The research was supported by the BioLOG project: supply side (both for virgin and reused materials), demand side, the authors are grateful for the support of National Center of 66 Science (NCN) through grant DEC-2020/39/I/HS4/03533, the [15] Benjamin Hague, Jana Kozáková, and Andrea Veselá. Closing the loop on Slovenian Research and Innovation Agency (ARIS) through wood – Circular Bioeconomy Opportunities in the Value Chain for Forest Products and Wood in Czechia . Tech. rep. Institute of Circular Economy grant N1-0223 and the Austrian Science Fund (FWF) through (INCIEN), 2023. grant I 5443-N. Balázs Dávid and Miklós Krész gratefully [16] Dmitry Ivanov. Structural Dynamics and Resilience in Supply Chain Risk acknowledge the Slovenian Research and Innovation Agency Management. Vol. 265. International Series in Operations Research & (ARIS) and the Ministry of the Economy, Tourism and Sport Management Science. Springer, 2018. (MGTŠ) for the grant V4-2512. Balázs Dávid is grateful for the [17] Dmitry Ivanov, Alexandre Dolgui, Boris Sokolov, and Marina Ivanova. “ Literature reviewon disruption recovery in the supply chain ” . In: support of the Slovenian Research and Innovation Agency International Journal of Production Research 55.20 (2017), pp. 6158–6174. (ARIS) through grant J1-50000. [18] Korina Katsaliaki, Panagiota Galetsi, and Sameer Kumar. “Supply chain disruptions and resilience: a major review and future research agenda”. In: References Annals of Operations Research 319 (2022), pp. 965–1002. [19] Nima Kazemi, Nikunja Mohan Modak, and Kannan Govindan. “A review [1] Heiner Ackermann, Erik Diessel, and Sven O. Krumke. “Robust flows with of reverse logistics and closed loop supply chain management studies adaptive mitigation”. In: EURO Journal on Computational Optimization 9 published in IJPR: a bibliometric and content analysis”. In: International [2] Mohammad Saeid Atabaki, Mohammad Mohammadi, and Bahman Naderi. [20] Alireza Khalili-Fard, Fatemeh Sabouhi, and Ali Bozorgi-Amiri. “Data-“ New robust optimization models for closed-loop supply chain of durable driven robust optimization for a sustainable steel supply chain network (2021), p. 100002. issn: 2192-4406. Journal of Production Research 57.15-16 (2019), pp. 4937–4960. [3] Engineering 146 (2020), p. 106520. Engineering 195 (2024), p. 110408. “ Alper Atamtürk and Muhong Zhang. Two-Stage Robust Network Flow [21] products: Towards a circular economy ” . In: Computers & Industrial design: Toward the circular economy ” . In: Computers & Industrial Kasper P.H. Lange, Gijsbert Korevaar, Inge F. Oskam, Igor Nikolic, and and Design Under Demand Uncertainty”. In: Operations Research 55.4 Paulien M. Herder. “Agent-based modelling and simulation for circular (2007), pp. 662–673. business model experimentation ” . In: Resources, Conservation & [4] Dimitris Bertsimas, Ebrahim Nasrabadi, and Sebastian Stiller. “Robust and Recycling Advances 12 (2021), p. 200055. Adaptive Network Flows”. In: Operations Research 61.5 (Oct. 2013), pp. [22] Haitham A. Mahmoud, Sarah Essam, Mohammed H. Hassan, and Arafa S. 1218–1242. Sobh. “ Modeling circular supply chains as an approach for waste [5] Dimitris Bertsimas and Melvyn Sim. “Robust discrete optimization and management: A systematic review and a conceptual framework ”. In: network flows”. In: Mathematical Programming 98.1–3 (Sept. 2003), pp. Journal of Engineering Research in press (2025). 49–71. [23] Yasmine Morjéne, Nadia Ndhaief, and Nidhal Rezg. “Optimization of [6] Dimitris Bertsimas and Melvyn Sim. “The Price of Robustness”. In: production batches in a circular supply chain under uncertainty”. In: IFAC Operations Research 52.1 (Feb. 2004), pp. 35–53. PapersOnLine 55.10 (2022), pp. 1752–1757. [7] Jacqueline Cramer. “The Raw Materials Transition in the Amsterdam [24] Gianfranco Pedone, József Váncza, and Ádám Szaller. “Exploring hidden Metropolitan Area: Added Value for the Economy,Well-Being, and the pathways to sustainable manufacturing for cyber-physical production Environment ” . In: Environment: Science and Policy for Sustainable systems”. In: Heliyon 10.8 (2024), e29004. issn: 2405-8440. Development 59.3 (2017), pp. 14–21. [25] Youngchul Shin, Gwang Kim, and Yoonjea Jeong. “Robust closed-loop [8] Rommert Dekker, Moritz Fleischmann, Karl Inderfurth, and Luk N. Van supply chain model with return management system for circular economy” Wassenhove, eds. Reverse Logistics: Quantitative Models for Closed-Loop . In: Computers & Industrial Engineering in press (2025) Supply Chains. Springer Berlin Heidelberg, 2004. doi: 10.1007/978- 3-540- [26] David Simchi-Levi, He Wang, and Yehua Wei. “Constraint Generation for 24803-3. url: https://doi.org/10.1007/978-3-540-24803-3. Two-Stage Robust Network Flow Problems”. In: INFORMS Journal on [9] Yann Disser and Jannik Matuschke. “The complexity of computing a robust Optimization 1.1 (Winter 2019), pp. 49–70. flow”. In: Operations Research Letters 48.1 (2020), pp. 18–23. [27] Emilia Taddei, Claudio Sassanelli, Paolo Rosa, and Sergio Terzi. “Circular [10] Péter Egri, Balázs Dávid, Tamás Kis, and Miklós Krész. “Robust facility supply chains theoretical gaps and practical barriers: A model to support location in reverse logistics”. In: Annals of Operations Research 324.1-2 approaching firms in the era of industry 4.0”. In: Computers & Industrial (2021). Engineering 190 (2024), p. 110049. issn: 0360-8352. [11] Péter Egri and Tamás Kis. “Robust two-stage optimisation in biomass [28] Tharaka De Vass, Alka Ashwini Nand, Ananya Bhattacharya, Daniel supply chains”. In: International Journal of Production Economics 285, Prajogo, Glen Croy, Amrik Sohal, and Kristian Rotaru. “Transitioning to a 109623 (2025) circular economy: lessons from the wood industry”. In: The International [12] Carmen Liping Fernández-Arribas, Borja Ponte, and Isabel Fernández. “ Journal of Logistics Management 34.3 (2023), pp. 582–610. Shaping closed-loop supply chain dynamics: Mitigating the bullwhip effect [29] Weiqi Yan, Nan Li, and Xin Zhang. “Enhancing supply chain management and improving customer satisfaction in production systems with material in the physical internet: a hybrid SAGA approach”. In: Scientific Reports reuse”. In: Computers & Industrial Engineering 195 (2024), p. 110407. 13.1 (2023), p. 21470. [13] Eilidh J. Forster, John R. Healey, Gary Newman, and David Styles. “ [30] Yuchen Zhao, Mohsen Roytvand Ghiasvand, and Babak Mohamadpour Circular wood use can accelerate global decarbonisation but requires cross- Tosarkani. “Balanced Uncertainty Sets for Closed-Loop Supply Chain sectoral coordination”. In: Nature Communications 14.6766 (2023). Design: A Data-Driven Robust Optimization Framework with Fairness [14] Supriyo Ghosh and Patrick Jaillet. “ An iterative security game for Considerations”. In: Expert Systems with Applications in press (2025) computing robust and adaptive network flows ” . In: Computers & Operations Research 138 (2022), p. 105558. issn: 0305-0548 67 Harvest plan generation in precision agriculture ∗ Štefan Horvat Damjan Strnad stef an.horvat@um.si Faculty of Electrical Engineering and Computer Science Faculty of Electrical Engineering and Computer Science Maribor, Slovenia Maribor, Slovenia damjan.strnad@um.si Domen Mongus Matej Brumen Faculty of Electrical Engineering and Computer Science Faculty of Electrical Engineering and Computer Science Maribor, Slovenia Maribor, Slovenia domen.mongus@um.si matej.brumen@um.si Abstract utilization of resources, such as agricultural equipment and labor. In this work, we present a method that utilises remote sensing One of many practical applications of precision agriculture is data, such as Sentinel-1 and Sentinel-2 imagery. Sentinel-1 uses planning and optimization of harvest times, which requires the synthetic aperture radar (SAR) and enables constant monitoring knowledge of crop maturity from which harvest time can be pre- regardless of the weather conditions and the time of day. The dicted. We present a simple procedure where we first obtain the resulting images usually contain a lot of backscatter that can Green Normalized Difference Vegetation Index (GNDVI) index be hard to interpret and analyse [3]. Sentinel-2 imagery is mul- values from Sentinel-2 imagery for a certain time period before tispectral and its main advantage over Sentinel-1 is the ability the crop enters the last growth cycle. The timeseries is analysed, to capture spectral reflectance, which is a lot more useful for and a simple linear and polynomial regression model are fitted. large scale crop analysis [4]. Most existing works focus on de- Extrapolation is used to calculate the intersection with the time tecting and monitoring specific growth cycles of crops to detect axis, which acts as the harvest date prediction. In the next step we and predict harvest dates using Sentinel-1. In [1, 9, 14] authors rearrange the fields based on crop maturity and create a harvest- analyse SAR backscatter signatures to detect crop growth cycles. ing plan that utilizes the combine harvester that maximizes the In [12, 17] machine learning models are first trained and than harvest area. We validated the results using actual harvest dates, used to predict the harvest dates. When using Sentinel-2 imagery and found that the polynomial regression results matched closer the most straightforward approach is to gather timeseries for an to the actual harvest dates than those of the linear regression. observed area and analyse the time series for any abrupt changes. This approach offers a scalable solution to harvest date prediction These can signify significant events happening on the field, which and plan generation, but further adjustments and a larger dataset is especially useful for post analysis [16]. are needed to improve the performance for practical application. In this paper, we present a simple algorithm for harvest time pre- Keywords diction and creation of a simple harvesting plan which simulates the anticipated harvest. Section 2 describes the methodology that precision agriculture, harvest maps, linear regression, polynomial was used in the approach, section 3 evaluates the results and regression section 4 concludes with potential future research directions. 1 Introduction 2 Methodology The development of Earth observation technologies, such as The process of harvest time optimization and harvest map gen- Sentinel-2 satellite network, remote sensing and GPS, has en- eration consists of two steps. In the first step, the field data is abled their widespread integration in to new areas. One of them gathered and predictions of optimal time for harvesting are made. is precision agriculture, which enables site specific interventions In the second step, an optimal harvest plan is created that aims to without physical presence and makes agriculture more sustain- maximize the combine harvester load based on the field maturity able, cost effective and increases the quality of the produce. This from the previous step. aligns well with the common agricultural policy (CAP) of the EU [2, 5, 13]. There are numerous interesting applications in preci- sion agriculture such as creating optimal fertilization plans [15], 2.1 Prediction of harvest times monitoring crop health and development [10], estimating yield The 𝐻 𝑎𝑟 𝑣 𝑒𝑠𝑡 𝑃 𝑟 𝑒𝑑𝑖𝑐𝑡 𝑖𝑜𝑛 algorithm accepts three inputs, a list of [17], creating harvest plans [11] and most recently the popular fields in GeoPackage (GPKG) format, which we denote as 𝐹 . The trend of artificial intelligence integration.[7, 17]. package contains metainformation about the fields, such as crop The problem of optimal harvest time deals with determining the type and field area. We only keep the fields that contain wheat exact time of crop maturity. It enables mass detection of fields or barley. The second input is the date (we will refer to it as or parts of fields that are ready or will be ready for harvest in the starting date), for which the harvest plan is being created. the near future. This enables efficient planning and maximum A 30-day window preceding the starting date is used to obtain the historic samples for each field in 𝐹 from Sentinel-2 data. All Permission to make digital or hard copies of all or part of this work for personal available Sentinel-2 images from the historic window are first or classroom use is granted without fee provided that copies are not made or cropped to field dimensions, and only images with cloud cover distributed for profit or commercial advantage and that copies bear this notice and lower than 15% are retained. For each field image, the GNDVI the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner /author(s). was calculated as: Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia © 2025 Copyright held by the owner/author(s). 𝑁 𝐼 𝑅 𝐺 − 𝐺 𝑁 𝐷𝑉 𝐼 (1) = 𝑁 𝐼 𝑅 𝐺 + 68 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Horvat et al. where NIR represents the near-infrared band and G represents where 𝑛 denotes the degree of the polynomial. After fitting the the green band. The GNDVI vegetation index is sensitive to polynomial to the data, the next step is to calculate, where the chlorophyll content, which helps at identifying late growth stages polynomial intersects the maturity threshold 𝑦 , which means 𝑓 [6]. The GNDVI values were clamped to be between 0 and 1. For that we need to solve the polynomial for that value and find the each valid timestamp, the mean of GNDVI over the field was real roots. The downside here is the need to choose the polyno- calculated and saved to a corresponding data structure. After mial degree and the right root values in case of multiple solutions. preprocessing, each field had a time series of mean GNDVI values, After experimenting with 𝑛 in the range from 2 to 4 we found 𝑛 of which were used in the forecasting step. An example of a GNDVI 2 to be sufficient. An example of fitting both models to a different time series can be seen in Figure 1 . GNDVI timeseries can be seeen in Figure 2. In the bottom Figure, the quadratic polynomial regression was used with two solutions Field 1239575, crop: Barley for the root. In our methodology, we select the solution that is GNDVI the nearest to the starting date. 0.62 Algorithm 1 Harvest plan generation 0.60 1: function HarvestPrediction(𝐹, 𝑆 𝑡 𝑎𝑟 𝑡 𝑖𝑛𝑔𝐷𝑎𝑡 𝑒, 𝑦 ) 𝑓 2: ⊲ 𝐹 : GeoPackage which contains a list of fields 0.58 3: ⊲ 𝑆 𝑡 𝑎𝑟 𝑡 𝑖𝑛𝑔𝐷𝑎𝑡 𝑒: Date for which the harvest plan will be created 0.56 4: ⊲ 𝑦 : Maturity threshold 𝑓 5: 0.54 6: 𝑛 ← 2 7: 𝑓𝑖 in 𝐹 for do 0.52 8: 𝑀 𝑒𝑡 𝑎𝐼 𝑛 𝑓 𝑜 ← GetMetainfo(𝑓 ) 𝑖 9: 𝑇 𝑆 GetSentinel2History(𝑆 𝑡 𝑎𝑟 𝑡 𝑖𝑛𝑔𝐷𝑎𝑡 𝑒) ← 2022-06-012022-06-032022-06-052022-06-072022-06-092022-06-112022-06-132022-06-152022-06-172022-06-19 10: 𝐿𝑖𝑛𝑅𝑒𝑔𝐼 𝑛𝑡 LinearRegression(𝑇 𝑆, 𝑦 ) ← 𝑓 11: 𝑃 𝑜𝑙𝑦𝑅𝑒𝑔𝐼 𝑛𝑡 PolyRegression(𝑇 𝑆, 𝑛, 𝑦 ) ← 𝑓 12: UpdateGeoPackageMetainfo(𝑓 ) 𝑖 Figure 1: Example of a GNDVI timeseries for a field with 13: end for barley. 14: end function 15: 16: HarvestPlan(𝐹 , 𝐻 𝑎𝑟 𝑣 𝑒𝑠𝑡 𝐷𝑎𝑡 𝑒, 𝑍 ) function To determine the optimal harvest time, we used two approaches. 17: ⊲ 𝐹 : GeoPackage which contains a list of fields The first approach is a simple linear regression, and the second 18: ⊲ 𝑍: Maximum area to harvest per day is a polynomial regression with the degree as a parameter. 19: ⊲ 𝐻 𝑎𝑟 𝑣 𝑒𝑠𝑡 𝐷𝑎𝑡 𝑒: Date when the harvest will begin 2.2 Predictions with linear regression 20: 21: SortByMaturity(𝐹 ) Because there exists significant correlation between GNDVI val- 22: 𝑡 𝑒𝑚𝑝 𝐷𝑎𝑡 𝑒 𝐻 𝑎𝑟 𝑣 𝑒𝑠𝑡 𝐷𝑎𝑡 𝑒 ← ues at consecutive timestamps, linear regression is the most natu- 23: 𝐹 <> empty while do ral approach. The dependent variable is the value of GNDVI, the 24: 𝑀 𝑒𝑡 𝑎𝐼 𝑛 𝑓 𝑜 GetMetainfo(𝑓 ) 𝑖 ← independent variable is time. The timestamps were converted to 25: 𝑄 CreateAndFillQueue(𝑡 𝑒𝑚𝑝 𝐷𝑎𝑡 𝑒,𝐹 ) ← POSIX format for easier handling of calculations. The relation- 26: 𝑡 𝑒𝑚𝑝 𝐷𝑎𝑡 𝑒 𝑡 𝑒𝑚𝑝 𝐷𝑎𝑡 𝑒 + 1 ← ship between dependent and independent variable can be defined 27: end while using the following formulation: 28: end function 𝑦 29: 𝑡 𝛽 0 𝛽 1 𝑥 𝑡 𝜖 𝑡 (2) 𝐹 = + + return where 𝑦𝑡 denotes the dependent variable, 𝑥𝑡 denotes the inde- pendent variable, coefficient 𝛽 0 denotes the intercept, 𝛽1 denotes the slope and 𝜖 𝑡 denotes the deviation from the fitted line [8]. 2.4 Creating the harvest plan The main application here is to determine the optimal time to The second step is to create an actual harvest plan that uses a harvest. Once the model is fitted, we can use extrapolation to single combine harvester for all fields in the list. The 𝐻 𝑎𝑟 𝑣 𝑒𝑠𝑡 𝑃 𝑙 𝑎𝑛 determine, where the regression line intersects the line of user- algorithm accepts three inputs. The first input is a GPKG of fields specified threshold value 𝑦 (which we will refer to as crop ma- with updated metainformation from the previous step, the second 𝑓 turity threshold): input is the harvest date, when the combine is going to start the 𝑦 actual harvesting, and the third is 𝑍, which denotes total harvest 𝛽 1 𝑓 − 𝑥 (3) = 𝑓 area a combine harvester can do every day. The fields in 𝐹 are 𝛽 0 first sorted by maturity. For every day the combine harvester is 2.3 Predictions with polynomial regression available, we add the fields from 𝐹 to a queue 𝑄 as long as their total area is smaller or equal to 𝑍 . The process can also tie-break Sometimes the changes in values between consecutive times- fields, so it is not necessary that the whole field is harvested in tamps exhibit a curved pattern. To capture such curvature in the one day. If fields remain in 𝐹 , we create new queue for the next data, we used a polynomial regression model, defined as: day and repeat the process until 𝐹 is empty. After assigning the 𝑦𝑡 𝛽0 𝛽1𝑥𝑡 𝛽2𝑥 𝛽 𝑡 = 2 𝑛 + + + ... 𝑛𝑥 (4) harvest dates to the fields, we store the fields back to the initial 𝑡 + 69 Harvest plan generation in precision agriculture Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Linear regression Table 1: Mean difference and std. deviation between pre- dicted and actual harvest times in days for linear regres- 0.5 sion. 0.4 Data Crop Mean Std. deviation Accuracy Value y=0.02 x 2022-07-16 0.3 Linear fit Barley 15.10 12.51 5/10 0.2 Wheat 23.10 20.79 2/8 0.1 0.0 Table 2: Mean difference and std. deviation between pre- dicted and actual harvest times in days for quadratic poly- nomial regression. 2022-06-01 2022-06-08 2022-06-15 2022-06-22 2022-07-01 2022-07-08 2022-07-15 Date Polynomial regression - Degree: 2 Crop Mean Std. deviation Accuracy 0.6 Data Barley 5.70 15.71 6/10 Poly fit Wheat 5.75 11.12 6/8 y=0.02 0.5 x 2022-01-16 x 2022-07-09 0.4 Value 0.3 0.2 Deviation from baseline (0) - Barley 0.1 30 0.0 25 2022-02 20 2022-03 2022-04 2022-05 2022-06 2022-07 Date 15 Figure 2: An example of fitting a linear model (top) and Deviation 10 a quadratic polynomial model (bottom) for a field with 5 wheat. Black dots mark the prediction which can be found 0 on the intersection between the maturity threshold 𝑦𝑓 = 0 2 4 6 8 0.02 and the time axis. Sample index 30 GeoPackage 𝐹 , populate the fields with new metainformation and 20 return the result to the user. The algorithms used in the process are described in 1. 10 Deviation 3 Results 0 A collection of eighteen fields located in northwestern Slovenia were used to evaluate the process. There were total of ten samples 10 with barley and eight samples with wheat. Sometimes the actual 0 2 4 6 8 harvests happened before or after the optimal time for harvesting Sample index because of bad weather or disruptions in resources (farming equipment breaks, organisational problems due to labor shortage). The algorithm was intended to be used near the start of the actual harvest season, which is usually determined by numerous Figure 3: Differences (in days) between predicted and actual exogenous factors. Our method assumes perfect conditions so it harvest dates on barley fields for linear (top) and quadratic does not take those limitations into account. polynomial (bottom) model. The vegetation index, the historic window length and maturity threshold 𝑦 were all chosen in conjunction with agricultural The comparison of harvest dates predicted with linear and 𝑓 experts. We should point out that the historic window can be in- polynomial regression compared to actual harvest dates can be creased, but then the preprocessing step takes more time. When seen in Tables 1 and 2. The polynomial regression has a lower doing extrapolation, both linear and quadratic polynomial re- average error than the linear regression for both crops, so its gression diverge far from the expected results or in the case of predictions were closer to the actual harvest dates in both cases. quadratic regression there can be no real solutions. To avoid this, Figures 3 and 4 show the residuals, i.e. the differences in days we clamp the harvest dates to the period between the 20th of between actual and predicted harvest days, for barley and wheat June and 31st of July, which are reasonable bounds for barley and fields, respectively. The opinion of agricultural experts is that wheat. all forecasts, that are within a ten day window from the actual 70 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Horvat et al. harvesting date can be considered acceptable, which is repre- no. 101084248 and Research Programme P2-0041. I would also sented with the accuracy column. The polynomial regression like to thank ITC for providing the actual harvest dataset. outperforms linear regression, as it scored an additional correct sample with barley and four additional samples with wheat. We 6 References can see that some predictions are off by multiple weeks. One [1] Arturo G. Cauba, Roshanak Darvishzadeh, Michael Schlund, Andrew Nelson, and Alice Laborte. 2025. Estimation of transplanting and harvest dates of rice of the reasons can be attributed to the small quantity of real crops in the philippines using sentinel-1 data. Remote Sensing Applications: samples. , 37, 101435. doi: https://doi.org/10.1016/j.rsase.202 Society and Environment 4.101435. 4 [2] European Commission. 1962. Retrieved August 21, 2025 from https://agricu Conclusion lture.ec.europa.eu/common- agricultural- policy_en. [3] ESA. 2014. Sentinel-1. Retrieved August 21, 2025 from https://dataspace.cop The described procedure is a simple and scalable solution for ernicus.eu/data- collections/sentinel- data/sentinel- 1. prediction of harvest dates and simple harvest plan generation. [4] ESA. 2015. Sentinel-2. Retrieved August 21, 2025 from https://www.esa.int It utilizes Sentinel-2 imagery to obtain a timeseries before the /Applications/Observing_the_Earth/Copernicus/Sentinel- 2. [5] Robin Gebbers and Viacheslav I. Adamchuk. 2010. Precision agriculture and nearing harvests and predicts harvest dates based on timeseries food security. , 327, 5967, 828–831. eprint: https://www.science.org Science patterns. We found that polynomial regression did a better job /doi/pdf /10.1126/science.1183899. doi: 10.1126/science.1183899. [6] Anatoly A. Gitelson, Yoram J. Kaufman, and Mark N. Merzlyak. 1996. Use at making those predictions and that the generated harvest plan of a green channel in remote sensing of global vegetation from eos-modis. can be organised as a simulation of how the harvest could be Remote Sensing of Environment , 58, 3, 289–298. doi: https://doi.org/10.1016 performed. The current method does not take into account real /S0034- 4257(96)00072- 7. [7] Garima Gupta. 2025. Applications of ai in precision agriculture. Discover world limitations, such as weather events, which could prove Agriculture. doi: 10.1007/s44279- 025- 00220- 9. vital if included in the harvest plan generation. Additional data [8] Rob Hyndman and G. Athanasopoulos. 2021. Forecasting: Principles and would also be of great importance because with a large enough . English. (3rd ed.). OTexts, Australia. Practice [9] Olena Kavats, Dmitriy Khramov, Kateryna Sergieieva, and Volodymyr Va- dataset a supervised machine learning model could be build that syliev. 2019. Monitoring harvesting by time series of sentinel-1 sar data. could predict the harvest dates with greater accuracy. , 11, 21. doi: 10.3390/rs11212496. Remote Sensing [10] James Y. Kim. 2024. Open-source software for satellite-based crop health monitoring. . doi: 10.1007/s42853- 024- 002 Journal of Biosystems Engineering 42- z. [11] Emogine Mamabolo, Makgabo Johanna Mashala, Ephias Mugari, Tlou Eliz- abeth Mogale, Norman Mathebula, Kabisheng Mabitsela, and Kwabena Deviation from baseline (0) - Wheat Kingsley Ayisi. 2025. Application of precision agriculture technologies for crop protection and soil health. Smart Agricultural Technology , 12, 101270. 60 doi: https://doi.org/10.1016/j.atech.2025.101270. [12] Gordan Mimić, {Amit Kumar} Mishra, Miljana Marković, Branislav Živalje- 50 vić, Dejan Pavlović, and Oskar Marko. 2025. Machine learning-based harvest date detection and prediction using sar data for the vojvodina region (ser- 40 bia). English. Sensors, 25, 7, (Apr. 2025). Publisher Copyright: © 2025 by the authors. doi: 10.3390/s25072239. Deviation agriculture: key advances and remaining knowledge gaps. Biosystems Engi-neering 20 30 [13] David J. Mulla. 2013. Twenty five years of remote sensing in precision , 114, 4, 358–371. Special Issue: Sensing Technologies for Sustainable Agriculture. doi: https://doi.org/10.1016/j.biosystemseng.2012.08.009. 10 [14] Ali Nasrallah, Nicolas Baghdadi, Mohammad El Hajj, Talal Darwish, Hatem Belhouchette, Ghaleb Faour, Salem Darwich, and Mario Mhawej. 2019. 0 Sentinel-1 data for winter wheat phenology monitoring and mapping. Re- mote Sensing 0 1 2 3 4 5 6 7 , 11, 19. doi: 10.3390/rs11192228. Sample index [15] Dorijan Radočaj, Mladen Jurišić, and Mateo Gašparović. 2022. The role of remote sensing data and methods in a modern approach to fertilization in 20 precision agriculture. , 14, 3. doi: 10.3390/rs14030778. Remote Sensing [16] Jan Verbesselt, Rob Hyndman, Achim Zeileis, and Darius Culvenor. 2010. Phenological change detection while accounting for abrupt and gradual 15 trends in satellite image time series. Remote Sensing of Environment, 114, 12, 2970–2980. doi: https://doi.org/10.1016/j.rse.2010.08.003. 10 [17] George Worrall, Jasmeet Judge, Kenneth Boote, and Anand Rangarajan. 2023. In-season crop phenology using remote sensing and model-guided Deviation 5 machine learning. Agronomy Journal, 115, 3, 1214–1236. eprint: https://acse ss.onlinelibrary.wiley.com/doi/pdf /10.1002/agj2.21230. doi: https://doi.org 0 /10.1002/agj2.21230. 5 0 1 2 3 4 5 6 7 Sample index Figure 4: Differences (in days) between predicted and actual harvest dates on wheat fields for linear (top) and quadratic polynomial (bottom) model. 5 Acknowledgments This research has received funding from the HORIZON Europe programme through project PrAEctiCe under grant agreement 71 Reducing #SAT to 𝑘-clique enumeration S´ andor Szab´ o Bogd´ an Zav´ alnij sszabo7@hotmail.com bogdan@renyi.hu University of P´ ecs HUN-REN Alfred Renyi Institute of Mathematics Pecs, Hungary Budapest, Hungary Abstract Complexity theory states that some complex problem can Figure 1: Φ = (𝑥1 ∨ 𝑥2 ∨ ¬𝑥3) ∧ (¬𝑥1 ∨ ¬𝑥2 ∨ 𝑥4) ∧ (𝑥2 ∨ be reduced to another problem. The reduction from SAT ¬𝑥 4 ∨ 𝑥3) ¬x 1 ¬x2 x4 to 𝑘-CLIQUE was already introduced by Karp. Here we propose an alternative graph reformulation, that can be efficiently used for solving the #SAT problem. Keywords x1 x2 model counting, combinatorial optimization, k-clique, SAT, reformulation x2 ¬x4 1 Introduction ¬x 3 x3 That is actually the base of complexity theory that some complex problem can be reduced to another problem. The reduction from SAT to 𝑘-CLIQUE was already introduced by Karp [3]. Thus most textbooks that include chapter proposed method is. This short paper is focusing only on on NP-completeness include reduction from 3-SAT to 𝑘- the #SAT problem. CLIQUE, as for example [2, pp. 1087–1089]. A small text- book like example pictured in Figure 1. Here the SAT 2 Proposed reformulation example consists of three clauses, each consisting of three We will describe a reduction of the SAT problem to the literals. Namely 𝑘-clique problem which differs from the commonly encoun- Φ = 𝐶1 ∧ 𝐶2 ∧ 𝐶3, tered reduction. The new 𝑘-clique reformulation of the 𝐶1 = 𝑥1 ∨ 𝑥2 ∨ ¬𝑥3, SAT problem has the added benefit that all the possible 𝐶2 solutions of the SAT problem and all the possible solutions = ¬ 𝑥 1 ∨ ¬ 𝑥2 ∨ 𝑥 4 , 𝐶3 = 𝑥2 ∨ ¬𝑥4 ∨ 𝑥3. Consequently, the of the 𝑘-clique problem are in a well defined connection. 𝑘-clique problem can be used to list all In short, we construct a graph. The number of the nodes is possible solutions of the SAT problem. equal to number of the literals in the SAT problem. Each Let 𝐶1, . . . , 𝐶𝑘 be clauses over the propositional variables node associated with a certain literal is not adjacent to 𝑥1, . . . , 𝑥𝑛. For our considerations we need a more detailed any node adjacent to a literal in the same clause; and two description of the clauses. Namely, for each 𝑖, 1 ≤ 𝑖 ≤ 𝑘 set nodes associated with literals 𝑙1, 𝑙2 from different clauses 𝐶𝑖 = 𝑙𝑖,1 ∨ 𝑙𝑖,2 ∨ · · · ∨ 𝑙𝑖,𝑠(𝑖), are not adjacent if 𝑙1 ∧ 𝑙2 is a contradiction. All other pairs are adjacent. The constructed graph is where 𝑙𝑖,1, 𝑙𝑖,2, . . . , 𝑙𝑖,𝑠(𝑖) are literals of the variables 𝑥1, . . . , 𝑥𝑛. 𝑘 -partite where 𝑘 is the number of clauses, and each As a first step we assign the expression 𝑘-clique is giving us a solution of the proposed SAT problem. 𝐿𝑖,𝑗 = 𝑙𝑖,𝑗 ∧ (¬𝑙𝑖,1 ∧ · · · ∧ ¬𝑙𝑖,𝑗−1) There are obviously several ways to accomplish such to the literal 𝑙𝑖,𝑗 for each 𝑖, 𝑗, 1 ≤ 𝑖 ≤ 𝑘, 1 ≤ 𝑗 ≤ 𝑠(𝑖). reformulation. We will introduce one, that may possibly For our previous small textbook like example the de- occurred elsewhere before. There is a small problem with scribed expressions will be the following: the textbook way, and that causes two different drawbacks. The problem is basically that two different 𝑘-cliques can 𝐿1,1 = 𝑥1, 𝐿1,2 = 𝑥2 ∧ ¬𝑥1, 𝐿1,3 = ¬𝑥3 ∧ ¬𝑥1 ∧ ¬𝑥2 . encode the same solution of the SAT problem. The reason 𝐿2,1 = ¬𝑥 , 𝐿 = ¬𝑥 ∧ 𝑥 , 𝐿 = 𝑥 ∧ 𝑥 ∧ 𝑥 . 1 2,2 2 1 2,3 4 1 2 behind another reformulation is twofold. First, for the SAT reformulation use a symmetry breaking and produce 𝐿3,1 = 𝑥2, 𝐿3,2 = ¬𝑥4 ∧ ¬𝑥2, 𝐿3,3 = 𝑥3 ∧ ¬𝑥2 ∧ 𝑥4. a graph with less edges and less 𝑘-cliques. Second, the Altogether nine 𝐿𝑖𝑗 formulas are associated with the textbook reformulation is hardly usable for #SAT but the SAT instance. We would like to point out that the 𝐿𝑖𝑗 formulas are not clauses but it will not cause any problem Permission to make digital or hard copies of all or part of this in the course of the clique reformulation of the problem. work for personal or classroom use is granted without fee provided So the equivalent formula for the original expression is: that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on Φ = 𝐾1 ∧ 𝐾2 ∧ 𝐾3, the first page. Copyrights for third-party components of this work ∨ 𝐿 ∨ 𝐿 , must be honored. For all other uses, contact the owner/author(s). 1 1,1 1,2 1,3 𝐾 = 𝐿 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia 𝐾 = 𝐿 ∨ 𝐿 ∨ 𝐿 , © 2 2,1 2,2 2,3 2024 Copyright held by the owner/author(s). 𝐾3 = 𝐿3,1 ∨ 𝐿3,2 ∨ 𝐿3,3 . 72 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Szab´ o et al. The reader can easily verify that the two formulas are in If there is a 𝑘-clique in the graph 𝐺, then Lemma 2.3. fact equivalent. We skip the proof, as we do use this fact there is an assignment of the truth values of the variables in our reasoning. 𝑥1, . . . , 𝑥𝑛 that makes the clauses 𝐶1, . . . , 𝐶𝑘 true simulta- As a second step we define a finite simple graph 𝐺. The neously. 𝑖 Proof. Let us suppose that the graph 𝐺 contains a , nodes of the graph 𝐺 are the conjunctions 𝐿𝑖,𝑗 for each 𝑗, 1 ≤ 𝑖 ≤ 𝑘, 1 ≤ 𝑗 ≤ 𝑠(𝑖). Two distinct nodes 𝐿𝑢,𝑣, 𝑘-clique ∆. We know that the nodes of the graph 𝐺 can 𝐿𝑝,𝑞 of the graph 𝐺 are not adjacent in 𝐺 if the formula 𝐿𝑢,𝑣 ∧ be wel-colored using 𝑘 colors. The nodes of the 𝑘-clique ∆ 𝐿 𝑝,𝑞 is a contradiction (always false). See as a small must receive 𝑘 pair-wise distinct colors. It means that the example the new graph for the same textbook problem we nodes of the 𝑘-clique ∆ receive the colors 1, 2, . . . , 𝑘. There described previously in Figure 2. are integers 𝑢(1), 𝑢(2), . . . , 𝑢(𝑘) such that 𝐿1,𝑢(1), 𝐿2,𝑢(2), . . . , 𝐿𝑘,𝑢(𝑘) (1) ∧(¬𝑥 1 ∨ (¬𝑥2 ∧ 𝑥1) ∨ (𝑥4 ∧ 𝑥1 ∧ 𝑥2)) Note that the expression 𝐿 𝑖,𝑢(𝑖) is true when each of the ∧ ( 𝑥 2 ∨ ( ¬ 𝑥 4 ∧ ¬ 𝑥 2 ) ∨ ( 𝑥 3 ∧ ¬ 𝑥 2 ∧ 𝑥 4 )) literals Figure 2: Φ = (𝑥1 ∨ (𝑥2 ∧ ¬𝑥1) ∨ (¬𝑥3 ∧ ¬𝑥1 ∧ ¬𝑥2 )) are the nodes of the 𝑘-clique ∆. ¬x1 ¬x2 ∧x1 x4 ∧x1 ∧ x2 𝑙𝑖,𝑢(𝑖), ¬𝑙𝑖,1, ¬𝑙𝑖,2, . . . , ¬𝑙𝑖,𝑢(𝑖)−1 is true. Using this information we can set the values of the corresponding propositional variables 𝑥1, . . . , 𝑥𝑛 to be true or false. Note, that it is not possible to assign true and false x1 x2 values at the same time to a given variable, since the nodes (1) pair-wise adjacent in the graph 𝐺. If a literal in the x2 ¬x4 expression 𝐿𝑢,𝑣 forces us to set the value of the variable 𝑥𝑖 ∧¬ x1 ∧¬x2 to be true and a literal in the expression 𝐿𝑥,𝑦 forces us to assign the variable 𝑥𝑖 the false value, then the expression ¬x3 x3 𝐿 ∧ 𝐿 contains 𝑥 ∧ ¬𝑥 and consequently 𝐿 ∧ 𝐿 𝑢,𝑣 𝑥,𝑦 𝑖 𝑖 𝑢,𝑣 𝑥,𝑦 ∧¬x1 ∧¬x2 ∧¬x2 ∧ x4 is a contradiction. This violates the fact that the nodes 𝐿 𝑢,𝑣, 𝐿𝑥,𝑦 of the graph 𝐺 are adjacent in the 𝑘-clique ∆. Of course, it well may happen that for some variables Lemma 2.1. The set of nodes 𝐿 {𝑖,1, . . . , 𝐿𝑖,𝑠(𝑖) } is an among 𝑥1, . . . , 𝑥𝑛 we are not forced to assign any of the independent set in the graph 𝐺 for each 𝑖, 1 ≤ 𝑖 ≤ 𝑘. This true or false values. In this case the value of such variable means that the nodes of the graph 𝐺 can be well-colored is not restricted and we are free to choose between the using 𝑘 colors, the nodes 𝐿𝑖,1 , . . . , 𝐿𝑖,𝑠(𝑖) all can receive truth or false values. In other words a 𝑘-clique in the graph color 𝑖. 𝐺 may give rise to more than one truth assignments of two distinct nodes Proof. 𝑧 In order to verify our observation let us consider the truth values of the variables that satisfy the clauses 𝐶 1 , . . . , 𝐶 𝑘 simultaneously. Namely there will be 2 of such 𝐿 𝑖,𝑝 , 𝐿 𝑖,𝑞 of the graph 𝐺 such that 𝑝 < 𝑞 . The formula assignments if the number of unsigned variables was 𝑧. 𝐿 𝑖,𝑝 contains the literal 𝑙 𝑖,𝑝 and the formula □ 𝐿𝑖,𝑞 contains the literal ¬𝑙𝑖,𝑝. Consequently, 𝐿𝑖,𝑝 ∧ 𝐿𝑖,𝑞 is a Remember, that in the graph 𝐺 two distinct nodes 𝐿𝑢,𝑣, contradiction. Therefore, the nodes 𝐿𝑖,𝑝, 𝐿𝑖,𝑞 of the graph 𝐿 𝑥,𝑦 are not adjacent if the formula 𝐿𝑢,𝑣 ∧ 𝐿𝑥,𝑦 is a con- 𝐺 are not adjacent in 𝐺. tradiction. Let 𝛾 be an assignment of truth values to the Lemma 2.2. If the clauses 𝐶1, . . . , 𝐶𝑘 can be satisfied propositional variables 𝑥1, . . . , 𝑥𝑛. Using 𝛾 we color cer- simultaneously, then the graph 𝐺 has a 𝑘-clique. tain edges of the graph 𝐺. The edge connecting the nodes 𝐿 𝑢,𝑣, 𝐿𝑥,𝑦 receives red color if the truth value assignment Proof. Let us assume that there is an assignment of the 𝛾 makes the expression 𝐿𝑢,𝑣 ∧ 𝐿𝑥,𝑦 true. If each edge of a truth values of the propositional variables 𝑥1 , . . . , 𝑥𝑛 that 𝑘-clique ∆ in the graph 𝐺 is red, then we call the 𝑘-clique makes each of the clauses 𝐶1, . . . , 𝐶𝑘 true. In particular ∆ a red 𝑘-clique in the graph 𝐺. We call the assignment for each 𝑖, 1 ≤ 𝑖 ≤ 𝑘 there is a literal 𝑙𝑖,𝑢(𝑖) such that the 𝛾 a satisfying assignment if 𝛾 makes each of the clauses assignment of the truth values makes 𝑙𝑖,𝑢(𝑖) true and makes 𝐶1, . . . , 𝐶𝑘 true simultaneously. Lemma 2.2 can can be 𝑙𝑖,1, . . . , 𝑙𝑖,𝑢(𝑖)−1 false. We simply pick the first among the restated in the following way. literals 𝑙𝑖,1, 𝑙𝑖,2, . . . , 𝑙𝑖,𝑠(𝑖) which is made true by the truth assignment of the variables. In the 𝑢(𝑖) = 1 particular case If 𝛾 is a satisfying assignment of the truth Lemma 2.4. the set of indices {1, . . . , 𝑢(𝑖) − 1} is empty. Obviously, this values of the variables 𝑥1, . . . , 𝑥𝑛, then there is exactly one assignment of the truth values of the variables makes the red 𝑘-clique in the graph 𝐺. conjunction Proof. Lemma 2.2 proved that there is at least one 𝐿𝑖,𝑢(𝑖) = 𝑙𝑖,𝑗 ∧ (¬𝑙𝑖,1 ∧ · · · ∧ ¬𝑙𝑖,𝑢(𝑖)−1) such clique, as it is constructed one. true for each Assume on the contrary that there are two non-identical 𝑖 , 1 ≤ 𝑖 ≤ 𝑘 . From this follows that 𝐿 𝑝,𝑢 ( 𝑝 ) ∧ 𝐿 red 𝑘-cliques ∆1 , ∆2 in the graph 𝐺. Since the red 𝑘-cliques 𝑞,𝑢 ( 𝑞 ) is true for each 𝑝 , 𝑞 , 1 ≤ 𝑝 < 𝑞 ≤ 𝑘 . In other words, the formula ∆ 1, ∆2 are not identical, there is a color class, say the 𝑖-th 𝐿 𝑝,𝑢 ( 𝑝 ) ∧ 𝐿 𝑞,𝑢 ( 𝑞 ) cannot be a contradiction. As a consequence the nodes color class, such that the nodes 𝐿𝑖,𝑝, 𝐿𝑖,𝑞 of ∆1, ∆2 in the 𝑖-th color class are not identical. 𝐿1,𝑢(1), 𝐿2,𝑢(2), . . . , 𝐿𝑘,𝑢(𝑘) As 𝐿 𝑖,𝑝 is an end point of a red edge in the red 𝑘- of the graph 𝐺 are the nodes of a 𝑘-clique in 𝐺. □ clique ∆1, the assignment 𝛾 makes the expression 𝐿𝑖,𝑝 73 Reducing #SAT to 𝑘-clique enumeration Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia true. Similarly, 𝛾 makes the expression 𝐿𝑖,𝑞 true. But from of size 922 996. After an edge dominance preconditioning Lemma 2.1 we know that 𝐿𝑖,𝑝 ∧ 𝐿𝑖,𝑞 is a contradiction, so the simplified graph had 8 233 edges, and the clique search 𝐿𝑖,𝑝, 𝐿𝑖,𝑞 cannot be true at the same time, so ∆1, ∆2 must lead to a search tree of size 72 810. be identical. □ It is clear from these examples, that both precondition- ing and the proposed reformulation may greatly aid the Theorem 2.5. The proposed reformulation along with calculation in some cases. the 𝑘-clique enumerating method, where for each clique We close this section with some clarifying notes. The one adds up numbers 𝑧 2 , where 𝑧 is the number of unset preconditioning rules used in connection with 𝑘-clique prob- variables, solves the #SAT problem. lems can be sorted into two broader classes. Namely, 𝑘- clique preserving and 𝑘-clique loosing rules. The 𝑘-clique Proof. Concludes from Lemmas 2.2,2.3 and 2.4. preserving rules reduces the given graph 𝐺 to a new graph 𝐺′ in such a way that each 𝑘-clique in 𝐺 is also a 𝑘-clique 3 Discussion ′ in the reduced graph 𝐺 . On the other hand, the 𝑘-clique One can notice, that if the ordering of the literals are fixed loosing rules reduces the given graph 𝐺 to a new graph in the same way in each clause then the proposed method ′′ 𝐺 such that some 𝑘-clique in 𝐺 may not be a 𝑘-clique in is very similar to the DPLL method adjusted for #SAT in ′′ the reduced graph 𝐺 but at least one of the 𝑘-cliques of [1]. The dedicated reader will realize that in the proposed the original graph 𝐺 still will be a 𝑘-clique in the reduced method rearranging the literals among each other in a ′′ graph 𝐺. Therefore, when our purpose is to enumerate clause will lead to essentially different graphs in the clique all the possible 𝑘-cliques of 𝐺, then we cannot simply ap- reformulation. These graphs not only have different number ply 𝑘-clique loosing preconditioning rules. However, when of edges, but the number of 𝑘-cliques can also differ. At our purpose is to verify that the given graph 𝐺 does not the first glance this may look as a disadvantage. However, have any 𝑘-clique, then we may apply 𝑘-clique loosing there is another way to look at this phenomenon. The sizes preconditioning rules. of the search spaces vary with permuting the literals in the The edge doninance preconditioning rule is a 𝑘-clique original SAT instance. Thus it opens up an avenue reducing loosing preconditiong rule. In the same time it was known the size of the search tree by tactically choosing among the that the examples from [4] are unsatisfiable. Therefore possible rearrangements of the literals. It gives a certain using 𝑘-clique loosing rules are fully justified in this case. flexibility, as one can use different methods (nowadays for On the other hand, surprisingly, even when there are example artificial intelligence) to find out which ordering several 𝑘-cliques present and the goal is to count them, and thus reformulation would give the best approach in one can use clique loosing preconditioning like dominance terms of solution speed. in the following way. In such a case, one need to keep a Also, one can apply preconditioning methods for the list of these transformations, and during the enumeration resulting graph based on graph theoretical considerations, each solution needs to be check against that list. This is like proposed in [5]. It may be much easier to enumerate justified by the fact that such transformations always delete all 𝑘-cliques in the resulting graph, this is a future goal of a node (edge) while pointing to another one as a possible our research. substitution. That is, if node 𝑢 dominates node 𝑣 and we We also did a tiny computational experiment. We took deleted node 𝑣, then during the solution listing process we two simple examples from [4], namely from Section 4, with need to check each solution if it contains (the remaining) parameters 𝑡 = 2, 3, where the SAT formula has 8𝑡 + 22 node 𝑢, and if substituting back node 𝑣 instead of node 𝑢 clauses and each of these clauses has length three, and the also gives a valid solution. formula is unsatisfiable, that is the number of solution is This approach however is not developed yet, and cer- zero. So our first 3-SAT example has 38 clauses, and the tainly needs a more elaborate program for listing solutions. second example has 46. The textbook reformulation of the first problem gave a References graph with 114 nodes and 6 218 edges. The clique search [1] Elazar Birnbaum and Eliezer L. Lozinskii. The good old davis- algorithm we used lead to a search tree of size putnam procedure helps counting models. Journal of Artificial 15 414 191 . Intelligence Research, 10(1):457–477, 1999. After an edge dominance preconditioning from [5] we got a [2] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, simplified graph of 5 967 edges, and the clique search lead and Clifford Stein. Introduction to Algorithms. The MIT Press, to a search tree of size third edition, 2009. 28 474 . The proposed reformulation [3] Richard M. Karp. Reducibility among combinatorial problems. of the first problem gave a graph with 114 nodes and In R. E. Miller, J. W. Thatcher, and J.D. Bohlinger, editors, 5 953 edges. The clique search algorithm we used lead to Complexity of Computer Computations, pages 85–103. New a search tree of size 92 296. After an edge dominance [4] Ming Ouyang. How good are branching rules in dpll? York: Plenum, 1972. Discrete preconditioning the simplified graph had 5 619 edges, and Applied Mathematics, 89(1):281–286, 1998. the clique search lead to a search tree of size [5] S´ andor Szab´ o and Bogd´ an Zav´ alnij. Clique search in graphs of 18 202 . special class and job shop scheduling. Mathematics, 10(5), 2022. The textbook reformulation of the second problem gave a graph with 138 nodes and 9 174 edges. The clique search algorithm lead to a search tree of size 2 497 099 319. After an edge dominance preconditioning we got a simplified graph of 8 730 edges, and the clique search lead to a search tree of size 142 399. The proposed reformulation of the sec- ond problem gave a graph with 138 nodes and 8 805 edges. The clique search algorithm we used lead to a search tree 74 Modularity aware graph clustering for exploratory tasks with a case study of the biomass supply chain Sylvert Prian Tahalea Arkadiusz Kawa Balázs Dávid sylvert@inf.u- szeged.hu Poznań School of Logistics, balazs.david@innorenew.eu University of Szeged, Hungary Department of Logistics, Poznań InnoRenew CoE, UP IAM and UP Universitas Pembangunan Nasional Poznań University of Economics and FAMNIT, University of Primorska, Veteran Yogyakarta, Indonesia Business, Department of Business Slovenia Relationships and International Marketing, Poznań Poland Abstract modularity [20, 17]. Modularity measurement objectively mea- sures the quality of the communities, where the value near one This paper proposes a novel modularity-aware graph clustering indicates well-structured communities, and a value near zero algorithm that combines label propagation principle and global indicates a bad community structure. There are several LPA ex- modularity. The algorithm consists of two phases: (1) the first tensions to increase the modularity utilizing its fast runtime, phase is to quickly form clusters using LPA and approximate namely LPAm [1], LPAm+ [13], LPA-MNI [12], and FLPA [18]. modularity to evaluate their quality, and (2) the second phase This paper proposes a novel modularity-aware graph clustering focuses on refining the cluster structures using global modula- algorithm utilising LPA, local modularity, and global modularity. tion. The proposed algorithm is later evaluated using several This algorithm consists of two phases: fast label propagation and metrics with several datasets and applied to a real-world use global modularity refinement. In the first phase, utilising LPA case. The results reveal a clear trade-off between internal and for the fast propagation and local modularity to regain its initial external cluster quality, which is useful for exploratory tasks. The structure. The second phase focuses on increasing the modular- modularity-aware graph clustering algorithm performed well in ity using global modularity refinement. Finally, the result will the experimental and the real-world cases, producing reliable provide higher-quality partitions compared to the original LPA. clusters for each case. The algorithm is later evaluated using different metrics with sev- Keywords eral datasets. Moreover, in order to invetsigate the quality of the methodology in real-world scenario, a use case of biomass supply graph clustering, modularity, LPA, exploratory data analysis chains is studied. 1 The remainder of the paper is organised as follows. Section 2 Introduction provides the methodology and evaluation techniques. Section 3 Graph clustering has arisen as a critical issue in network research, Sec- provides the main results and highlights the main findings. data mining, and machine learning due to its potential to reveal tion 4 Section 5 provides the use case on a real-world scenario. hidden structures within complex systems. Real-world phenom- concludes the contribution, limitations, and future research. ena such as social interactions, biological processes, communica- tion networks, citation graphs, and transportation systems can 2 Methodology be naturally modeled as graphs, with nodes representing things and edges representing their relationships. Identifying cohesive 2.1 Modularity groups of nodes, also known as clusters or communities, allows Modularity is one of the most influential principles in graph clus- researchers to get a better understanding of the modular organi- tering, which is evaluated by the strength of a network’s partition sation of networks and simplify large-scale data for analysis. by comparing the density of edges inside communities [15]. High Over the past decades, various algorithmic paradigms have arisen, modularity value indicates that clusters have significantly more ranging from modularity optimisation, spectral approach, and internal connections than external ones. Modularity optimisa- random walks to label propagation, statistical inference, and most tion has become a benchmark for measuring cluster quality, but recently, utilising graph neural networks. Each technique strikes the direct optimisation of modularity is challenging due to its a balance between accuracy, scalability, interpretability, and ap- NP-hard nature. The modularity measurement [15] is as follows. plicability to overlapping or hierarchical communities. Despite 1 ∑︁ 𝑘 𝑘 𝑖𝑗 these achievements, there are some problems remaining, such as 𝑄 𝐴𝑖 𝑗 𝛿 𝑐𝑖 , 𝑐𝑗 , (1) = [ − ] ( ) 2𝑚 2𝑚 the low modularity for the fast algorithms, efficient methods for large-scale graphs, adaptive strategies for temporal graphs, and where 𝐴𝑖 𝑗 represents the actual connection between nodes 𝑖 and the lack of universally acknowledged ground truth. 𝑘 𝑘 𝑖 𝑗 𝑗 , the term is the expected number of edges between nodes 2𝑚 The Label propagation algorithm (LPA) is considered to be one 𝑖 and 𝑗, and 𝑐𝑖 is the community assignment. of the fastest graph clustering techniques. However, it has low To avoid recomputing modularity for every possible partition, Permission to make digital or hard copies of all or part of this work for personal Blondel et al. [2] proposed a local measurement that evaluates the or classroom use is granted without fee provided that copies are not made or modularity change if a node were moved from its original cluster distributed for profit or commercial advantage and that copies bear this notice and to the neighboring cluster and also provides the approximate the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner /author(s). modularity. The approximate modularity is as follows. Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia © 2024 Copyright held by the owner/author(s). 𝑘 𝑘 sum_tot 𝑙 𝑣,𝑖𝑛 𝑣 · () Δ 𝑄 𝑣 , 𝑙 (2) ( ) ≈ − 2 𝑚 2𝑚 75 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Tahalea et al. Algorithm 1 Fast propagation phase Algorithm 2 Global Modularity Refinement Require: Graph 𝐺 = (𝑉 , 𝐸) 1: Input: Label map 𝐿, Graph 𝐺, stability threshold 𝑅 Ensure: Node labels 𝐿(𝑣) for all 𝑣 ∈ 𝑉 2: Initialize stability counter 𝑆 (𝑣) ← 0 for all 1: Assign each node 𝑣 𝑉 a unique label 𝐿 𝑣 𝑣 3: ∈ ( ) ← repeat 2: 4: nodes 𝑣 𝑉 not marked stable repeat for all ∈ do 3: nodes 𝑣 𝑉 in random order 5: Compute label frequencies among neighbors for all ∈ do 4: Compute label frequencies among neighbors of 𝑣 6: Select Top-𝑘 candidate labels (optional) 5: Select top-𝑘 frequent labels (optional) 7: labels 𝑙 for all do 6: candidate labels 𝑙 8: Temporarily assign 𝐿 𝑣 𝑙 do ( ) ← for all ′ 7: Estimate local modularity gain 𝑄 𝑣, 𝑙 9: Compute new global modularity𝑄 Δ ( ′ ) 8: 10: end for end for 9: ∗ ∗ ′ 11: Select label 𝑙 with maximum gain (if gain > threshold) Select label 𝑙 that maximizes 𝑄 10: ∗ ′ 12: Update 𝐿 ( 𝑣 ) ← 𝑙 if 𝑄 > 𝑄 then 11: 13: Update 𝐿 𝑣 𝑙 , 𝑆 𝑣 0 ( ) ← ( ) ← end for ∗ 12: no label changes or maximum iterations reached 14: until else 15: 𝑆 𝑣 𝑆 𝑣 1 ( ) ← ( ) + 16: 𝑆 𝑣 𝑅 if ( ) ≥ then 17: Mark 𝑣 as stable where 𝑘 is the number of edges from node 𝑣 carrying label 𝑙; 18: end if 𝑣,𝑖𝑛 19: end if 𝑘𝑣 is the degree of node 𝑣; sum_tot(𝑙) is the total degree of nodes 20: end for with label (𝑙); and 𝑚 is the number of edges in the network. until 21: no label changes or improvement < 𝜖 2.2 Label Propagation Algorithm The label propagation algorithm (LPA) was introduced as one 2.4 Evaluation of the simplest yet most scalable methods for graph clustering The proposed algorithm is evaluated using several common met- [16]. The algorithm begins by assigning a unique label to each rics that are widely accepted for graph clustering, such as the node, then iteratively updates the node’s label based on the most modularity measurement, normalized mutual information (NMI), frequent label among its neighbors. The process is performed Adjusted Rand Index (ARI), and F1-score. The modularity mea- in random sequential order to avoid bias, and a label is chosen surement measures the internal structures of the cluster using at random when there are multiple labels that have the same Equation (1), while NMI measures the similarity between the frequency. The iterations continue until every node holds a label result cluster and the ground truth of the dataset as follows [4]. that is the majority among the neighbors, leading to convergence 2 𝐼 𝐴, 𝐵 · () in the network. The clusters are formed by grouping the nodes 𝑁 𝑀 𝐼 𝐴, 𝐵 (3) ( ) = 𝐻 𝐴 𝐻 𝐵 ( ) + () that share the same label. LPA has near-linear time complex- where 𝐼 𝐴, 𝐵 is the mutual information between clusters 𝐴 and ( ) ity, ability to uncover clusters without prior knowledge of their 𝐵; and 𝐻 𝐴 and 𝐻 𝐵 are the entropies of the clusters. ( ) ( ) number, and is scalable for large networks. The Adjusted Rand Index (ARI) is used to measure similarity 2.3 Modularity-Aware Graph Clustering between generated clusters and ground truth [6], while F1-score evaluates the precision and recall for each cluster [14, 3]. The The modularity-aware graph clustering utilizes LPA’s ability to ARI is measured using Equation (4) and F1-score is measured quickly cluster the nodes and modularity refinement to produce using Equation (5). better cluster structures. This proposed method is divided into h i two phases: (1) The first phase is to quickly form clusters using Í 𝑛𝑖 𝑗 Í 𝑎 Í 𝑏 𝑛 𝑖 𝑗 − / 𝑖 𝑗 2 𝑖 2 𝑗 2 2 LPA and approximate modularity to evaluate their quality, and ARI (4) = h i h i (2) the second phase focuses on refining the cluster structures 1 Í 𝑎𝑖 𝑏 Í 𝑎 𝑏 𝑖 Í 𝑛 𝑗 Í 𝑗 + − / 2 𝑖 2 𝑗 2 𝑖 2 𝑗 2 2 using global modularity measurement. where 𝑛 is the number of nodes; 𝑛 is the number of nodes in 𝑖, 𝑗 2.3.1 Phase 1: Fast Propagation Phase. The objective of the first both the predicted cluster 𝑖 and the ground truth cluster 𝑗; 𝑎𝑖 is phase is to obtain the local approximation of the modularity gain the number of nodes in the predicted cluster 𝑖; 𝑏 is the number 𝑗 by rapidly propagating through the network. This step helps to of nodes in the ground truth cluster 𝑥 𝑗 ; and is a binomial coef- 2 form a coarse graph for further refinement. The initial label for ficient to count the number of pairs between clusters. each node is its own node ID; then the algorithm iterates over all the nodes in a randomized order and updates the labels using 2 precision recall 2 𝑇 𝑃 · · · 𝐹1 (5) = = the local approximation of modularity gain as in Equation (2). precision recall 2 𝑇 𝑃 𝐹 𝑃 𝐹 𝑁 + · + + This step also leverages the Top-K label filtering to reduce the where 𝑇 𝑃 are correctly predicted positive cases; 𝐹 𝑃 are incor- chance of unstable changes while a minimum gain threshold is rectly predicted as positive; and 𝐹 𝑁 are missed positive cases, applied to prevent weak label changing. The iteration process which are predicted as negative. will continue until there is no label change or the maximum number of iterations is reached. The fast propagation algorithm is presented in Algorithm 1. 3 Experiments and results 2.3.2 Phase 2: Modularity Refinement. . The objective of the sec- ond phase is to refine the approximation performed in the first phase. Whenever the label is spread across the network, maxi- 3.1 Experimental Datasets and results mal modularity is gained from the overall structure. Therefore, There are four real-world datasets used in this research, such as the second phase performs the refinement for each label using Zachary’s Karate Club Network [5], Football [5], Polbooks [8], global modularity, as in Equation (1), for the entire network. The and Email [19, 10, 11] from SNAP datasets. The synthetic datasets modularity refinement is presented in Algorithm 2. are also used in this research, utilizing the LFR framework [9]. 76 Modularity aware graph clustering Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia The summary of the datasets is presented in Table 1, where Table 2: Comparison of algorithms in terms of modularity 𝑐 is the number of clusters, 𝑘 is the average degree, 𝜇 is the and NMI across datasets mixing parameter. The measurement is compared with several algorithms such as LPA, LPAm+, and Louvain. This is meant to Dataset Modularity compare the proposed algorithm with its predecessors and one LPA LPAm+ MA-LPA Louvain of the top graph clustering algorithms. Karate 0.309 0.418 0.427 0.445 Polbook 0.481 0.493 0.519 0.528 Table 1: Comparative metadata of benchmark datasets Football 0.583 0.557 0.604 0.596 Email 0.089 0.418 0.431 0.432 LFR1 0.807 0.809 0.808 0.810 Dataset Nodes Edges 𝑐 𝑘 𝜇 LFR2 0.479 0.514 0.511 0.515 Karate 34 78 2 - - 0.742 LFR3 0.569 0.735 0.731 Football 115 613 3 - - 0.830 0.830 LFR4 0.829 0.824 Polbooks 105 441 12 - - 0.841 0.841 LFR5 0.839 0.840 Email 1005 25571 42 - - 0.613 0.613 LFR6 0.611 0.609 LFR1 1000 10455 44 20.91 0.10 NMI LFR2 2000 15495 32 15.49 0.15 LPA LPAm+ MA-LPA Louvain LFR3 5000 93743 56 37.50 0.30 1.000 Karate 0.565 0.607 0.483 LFR4 10000 113290 61 22.66 0.10 1.000 Polbooks 0.611 0.759 0.697 LFR5 10000 120047 75 24.01 0.20 1.000 Football 0.919 0.937 0.939 LFR6 10000 190440 84 38.09 0.25 Email 0.180 0.593 0.575 0.663 LFR1 0.932 0.964 0.993 1.000 LFR2 0.831 0.854 0.823 1.000 3.1.1 Modularity and NMI. The modularity evaluation is meant LFR3 0.301 0.602 0.615 0.598 to evaluate the structure of the clusters. The larger value, closer 1.000 LFR4 0.986 0.989 0.964 to 1, indicates a densely connected structure within the results, LFR5 0.979 0.994 0.995 1.000 but it does not indicate the accuracy of the community detection. LFR6 0.961 0.975 0.963 1.000 The results presented in Table 2 show that the modularity aware label propagation algorithm has the best performance across the datasets, from the small one to large datasets. The modularity Table 3: Comparison of algorithms in terms of ARI and aware label propagation performed at the same level as Louvain F1-score across datasets in the LFR4-LFR6 with the highest modularity score, showing that it can produce high-quality clusters. Dataset ARI The normalized mutual information measures the similarity be- LPA LPAm+ MA-LPA Louvain tween clusters generated by the algorithms and the ground truth. Karate 0.398 0.597 0.483 1.000 Based on the results presented in Table 2, the LPA is the best Polbooks 0.509 0.710 0.579 1.000 performer, reaching the highest score for almost all the datasets. 1.000 Football 0.833 0.842 0.892 The modularity aware label propagation algorithm is the second 0.440 Email 0.011 0.332 0.315 best for NMI evaluation, with notably one highest score for the 1.000 LFR1 0.778 0.905 0.976 Email network dataset, having a huge gap with LPA, which is 1.000 LFR2 0.400 0.411 0.387 the best overall NMI performer. 0.487 LFR3 0.177 0.471 0.462 3.1.2 ARI and F1-score. LFR4 The Adjusted Rand Index (ARI) quanti- 1.000 0.911 0.966 0.959 LFR5 0.955 0.980 0.979 1.000 fies the similarity between the generated clusters and the ground LFR6 0.918 0.899 0.903 1.000 truth, considering all pairs of samples and evaluating whether they are assigned to the same cluster. The results presented in F1-score Table 3 show that only LPA is the best performer. The modularity LPA LPAm+ MA-LPA Louvain aware label propagation only has the highest score once, for the Karate 0.807 0.956 0.850 1.000 Email network case, while becoming the second best for most Polbooks 0.688 0.897 0.665 1.000 other cases. Football 0.869 0.986 0.745 1.000 The F1-score serves as an evaluation metric for assessing the Email 0.078 0.128 0.130 0.261 quality of cluster assignments. The results presented in Table LFR1 0.546 0.710 1.000 1.000 3 show that LPA produced high-quality clusters, reaching the LFR2 0.907 0.998 0.855 1.000 highest score for most of the real-world and synthetic datasets. 0.512 LFR3 0.185 0.491 0.498 The modularity-aware label propagation algorithm performs as 1.000 1.000 LFR4 0.661 0.877 the second best and joins the LPA as the best performer for the 1.000 1.000 LFR5 0.759 0.0.993 three largest datasets in this research. 1.000 1.000 LFR6 0.796 0.837 3.2 Discussion The results highlight a clear distinction between internal and 2, indicating densely connected clusters in the graph topology. external clustering quality. The modularity-aware label propaga- However, the standard LPA attains the strongest argument with tion consistently maximizes the modularity, as presented in Table ground truth across most datasets, suggesting that it works better 77 Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Tahalea et al. for datasets with known labels. The ARI and F1-score results in the data, partitioning the firms into clusters without any apriori Table 3 show the consistent trade-offs of modularity-aware label categorization. However, the results demonstrate that explicitly propagation, which is optimizing the global structural objective optimizing for modularity yields a superior segregation of the not always gives the best intended partition against known la- network. bels. Although no explicit categories (such as firm type or size) were This means that the algorithm choice should reflect the evalu- given to the clustering algorithms, the communities they de- ation goal. When the task is exploratory and the ground truth tected appear to reflect meaningful behavioral and operational is not available, maximizing internal structure is attractive for patterns among the companies. In other words, firms ended up discovering cohesive clusters; thus, modularity-aware label prop- clustered together because they answered many survey questions agation algorithms are better. When the aim is to recover known in similar ways – a purely data-driven outcome that likely cor- categories or enable downstream tasks, LPA is safer given its responds to real-world commonalities. For example, one cluster superior NMI, ARI, and F1-score. derived from the graph predominantly consists of sawmills and wood processors that generate large volumes of residues and use 4 Case study them for bioenergy, while another cluster groups manufacturers 4.0.1 (e.g. furniture or flooring producers) that have different residue Dataset. This case study draws on a quantitative survey uses and logistics practices. Indeed, the analysis suggests that conducted in late 2022 within the BioLog project, aimed at de- companies naturally form a few distinct sub-communities: even veloping a model of the reverse supply chain of residual wood without predefining any segments, those with analogous supply biomass. The survey targeted enterprises across the wood sec- chain roles, residue utilization strategies, or challenges tend to tor—sawmills, furniture and joinery producers, recyclers, and congregate in the same community. This insight is valuable for logistics providers—using a stratified random sampling method policymakers and industry stakeholders – it implies the sector based on PKD (The Polish Classification of Activities) industry can be segmented into groups with shared characteristics, which codes and a project-specific list. Data were collected through may each benefit from tailored strategies (for instance, a cluster 300 Computer Assisted Telephone Interviews (CATI) with com- of firms focused on energy production might face similar regula- pany representatives knowledgeable about wood residue origins tory and technological issues). and management. The questionnaire gathered company profile In constructing the similarity measure, all survey features were data (value chain position, employment size, revenue) and ex- treated with equal weight. This methodological choice (i.e. not plored types of wood residues used (forestry/agricultural, post- assigning higher importance to any particular question or topic) production, post-consumer) as well as their applications in energy means the clustering was driven by overall similarity across generation, energy carriers, and wood-based materials. It also many attributes. A side effect is that very common attributes investigated logistics and technological processes, cooperation (such as broadly adopted practices) contribute to linking many networks, requirements, and constraints in biomass management, firms, potentially overshadowing rarer but distinctive features. and adaptations in transport, storage, and unitization practices. In the present case, however, even unweighted features yielded The dataset was anonymised, and respondents were assured of intelligible clusters, indicating that the dominant patterns in the confidentiality, informed consent, and voluntary participation. data were strong enough to shape the communities. Future work 4.0.2 Results. The dataset underwent a series of preprocessing could experiment with feature weighting (for instance, giving steps before being analyzed. The answers to the questionnaire more emphasis to specific key questions or rare responses) to see were transformed into categorical and numerical data. The In- if even clearer or more nuanced groupings emerge. Nonetheless, terquartile Range (IQR) performed to detect the outlier data and the current graph clustering results already highlight consistent resulted in reducing the number of questions to only include patterns: companies with similar operational profiles gravitated relevant ones. The clean data was used to model a graph with into the same clusters. In summary, the modularity-aware graph the nodes representing the firms and the edge weight between clustering of the BioLog survey data uncovers a modular struc- any two companies using position-aware Jaccard similarity [7]: ture in the wood biomass supply chain, revealing that despite essentially measuring the fraction of survey questions on which the lack of explicit grouping criteria, firms naturally aggregate the two firms gave identical answers. There were several versions into network communities that mirror their shared behaviors of graph models such as unweighted graph, plain weighted graph, and challenges. The performance of the algorithms is presented and weighted graph with thresholds (0.1, 0.125, and 0.25) in Table 4. The graph indicates that most businesses are connected through shared practices. Out of 300 firms, 272 (about 91%) form one large connected group, while the other 28 are isolated or in very small clusters, representing outliers. This core network isn’t very dense, 5 Conclusion but it does have clusters: companies that work with the same In conclusion, the results reveal a clear trade-off between internal partner are commonly connected to each other, creating tiny and external cluster quality; the modularity-aware label propaga- groupings with shared qualities. The average distance between tion consistently attains the highest modularity, which is useful companies is modest (approximately 3 steps), and the longest for exploratory tasks. The standard LPA achieved the best agree- distance is tiny (6–7 steps), which gives the network a "small- ment with ground truth on NMI, ARI, and F1-score across most world" shape. In practice, this means that any two firms, even real and synthetic datasets, showing it’s better for optimizing from different parts of the sector, can be connected through only downstream tasks. Overall, these findings argue against a single a few intermediaries. best method and support choosing the algorithm according to the The proposed algorithm applied to this data, alongside with Lou- evaluation objective and data regime. Moreover, a real-world case vain, LPA, and LPAm+ to identify the clusters. The graph clus- study of biomass supply chains demonstrates that our methodol- tering methods uncovered a non-trivial clustering structure in ogy provides results for strategic decision making. 78 Modularity aware graph clustering Information Society 2024, 7–11 October 2024, Ljubljana, Slovenia Table 4: Algorithms performance in the case study [8] Valdis Krebs. 2004. Books about us politics. unpublished, http://www. orgnet. com. Unweighted Graph [9] Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi. 2008. Bench- mark graphs for testing community detection algorithms. Physical Review LPA LPAm+ MA-LPA Louvain , 78, 4, 046110. doi: 10.1103 E—Statistical, Nonlinear, and Soft Matter Physics No. of Cluster(s) 1 1 1 3 /PhysRevE.78.046110. [10] Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. 2007. Graph evolution: Modularity 0 0 0 densification and shrinking diameters. ACM transactions on Knowledge 0.259 Weighted Graph Discovery from Data (TKDD), 1, 1, 2–es. doi: 10.1145/1217299.1217301. [11] Jure Leskovec and Andrej Krevl. 2014. SNAP Datasets: Stanford large net- LPA LPAm+ MA-LPA Louvain work dataset collection. http://snap.stanf ord.edu/data. (2014). No. of Cluster(s) 1 1 6 [12] Huan Li, Ruisheng Zhang, Zhili Zhao, and Xin Liu. 2021. Lpa-mni: an im-8 Modularity 0 0 0.145 0.147 proved label propagation algorithm based on modularity and node impor- tance for community detection. , 23, 5, 497. doi: 10.3390/e23050497. Entropy Weighted Graph (0.1) [13] Xin Liu and Tsuyoshi Murata. 2010. Advanced modularity-specialized label LPA propagation algorithm for detecting communities in networks. LPAm+ MA-LPA Louvain Physica A: Statistical Mechanics and its Applications, 389, 7, 1493–1500. doi: 10.1016/j.p No. of Cluster(s) 1 1 8 10 hysa.2009.12.019. Modularity 0 0 0.139 [14] Christopher D Manning. 2008. . Syngress 0.155 Introduction to information retrieval Weighted Graph (0.125) Publishing. [15] Mark EJ Newman and Michelle Girvan. 2004. Finding and evaluating com- LPA LPAm+ MA-LPA Louvain munity structure in networks. Physical review E, 69, 2, 026113. doi: 10.1103 No. of Cluster(s) 1 1 10 10 /PhysRevE.69.026113. [16] Usha Nandini Raghavan, Réka Albert, and Soundar Kumara. 2007. Near Modularity 1 1 0.146 0.153 linear time algorithm to detect community structures in large-scale net- Weighted Graph (0.25) works. , 76, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 3, 036106. doi: 10.1103/PhysRevE.76.036106. LPA LPAm+ MA-LPA Louvain [17] Seema Rani and Monica Mehrotra. 2017. Hybrid influential centrality based No. of Cluster(s) 1 1 4 label propagation algorithm for community detection. In 10 2017 International Modularity Conference on Computing, Communication and Automation (ICCCA). IEEE, 0 0 0.266 0.238 11–16. doi: 10.1109/CCAA.2017.8229801. [18] Vincent A Traag and Lovro Šubelj. 2023. Large network community detec- tion by fast label propagation. , 13, 1, 2701. doi: 10.1038/s4 Scientific Reports Finally, there are several methodological considerations that need 1598- 023- 29610- z. [19] Hao Yin, Austin R Benson, Jure Leskovec, and David F Gleich. 2017. Local to be improved. Future work could explore several improvements, higher-order graph clustering. In Proceedings of the 23rd ACM SIGKDD such as (1) multi-objective formulation that balances modularity international conference on knowledge discovery and data mining, 555–564. with information-theoretic alignment; (2) considering variability doi: 10.1145/3097983.3098069. [20] Aiping Zhang, Guang Ren, Yejin Lin, Baozhu Jia, Hui Cao, Jundong Zhang, over multiple runs and using consensus clustering; and (3) exam- and Shubin Zhang. 2014. Detecting community structures in networks by ining performance under different mixing parameters and larger label propagation with prediction of percolation transition. The Scientific datasets. World Journal, 2014, 1, 148686. doi: 10.1155/2014/148686. Acknowledgements The research was supported by the BioLOG project: the authors are grateful for the support of the National Center of Science (NCN) through grant DEC-2020/39/I/HS4/03533, the Slovenian Research and Innovation Agency (ARIS) through grant N1-0223, and the Austrian Science Fund (FWF) through grant I 5443-N. Balázs Dávid is also grateful for the support of the Slovenian Research and Innovation Agency (ARIS) through grant J1-50000, and gratefully acknowledges the Slovenian Research and Innova- tion Agency (ARIS) and the Ministry of the Economy, Tourism and Sport (MGTŠ) for the grant V4-2512. References [1] Michael J Barber and John W Clark. 2009. Detecting network communities by propagating labels under constraints. Physical Review E—Statistical, Non- linear, and Soft Matter Physics, 80, 2, 026129. doi: 10.1103/PhysRevE.80.0261 29. [2] Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. 2008. Fast unfolding of communities in large networks. Journal of statistical mechanics: theory and experiment, 2008, 10, P10008. doi: 10.1088/1 742- 5468/2008/10/P10008. [3] Peter Christen, David J Hand, and Nishadi Kirielle. 2023. A review of the f- measure: its history, properties, criticism, and alternatives. ACM Computing Surveys, 56, 3, 1–24. [4] Leon Danon, Albert Diaz-Guilera, Jordi Duch, and Alex Arenas. 2005. Com- paring community structure identification. Journal of statistical mechanics: Theory and experiment, 2005, 09, P09008. doi: 10.1088/1742- 5468/2005/09 /P09008. [5] Linton C Freeman. 1977. A set of measures of centrality based on between- ness. , 35–41. doi: 10.2307/3033543. Sociometry [6] Lawrence Hubert and Phipps Arabie. 1985. Comparing partitions. Journal of classification, 2, 1, 193–218. doi: 10.1007/BF01908075. [7] Paul Jaccard. 1901. Étude comparative de la distribution florale dans une portion des alpes et des jura. , 37, 547–579. Bull Soc Vaudoise Sci Nat 79 A Synthetic Multi-View Tracking and 3D Pose Dataset for Automated Airport Visual Surveillance Ahmed Mansour Nadezda Kirillova Csaba Beleznai Horst Possegger Fabio F. Oberweger Institute of Visual Computing Verena Widhalm Graz University of Technology firstname.lastname@ait.ac.at Graz, Austria Assistive & Autonomous Systems, AIT Austrian Institute firstname.lastname@tugraz.at of Technology GmbH Vienna, Austria ABSTRACT Multiple object tracking (MOT) is a core task of automated vi- sual surveillance systems. Driven by Deep Learning, recent ad- vances have significantly improved tracking accuracy and robust- ness across varied scenarios. However, wide-area visual surveil- lance, such as observing airport ground and airborne objects, still remains challenging. In such scenarios the presence of small ob- jects, numerous visually similar targets, frequent occlusions, atmo- spheric effects, scale diversity, and substantial scene clutter con- tinue to degrade detection and tracking performance. To address these limitations, we present the AirTrackSim25 synthetic airport surveillance dataset, designed to support and advance research in object detection, tracking, and data association. The dataset includes multiple wide-area perspectives of an airport environ- ment, featuring rich structural details and realistic simulations of ground and air-based object motion. Each aircraft is annotated with comprehensive 2D and 3D bounding box and key-point in- Figure 1: Visual summary of the main dataset characteristics, formation, as well as motion trajectories. We demonstrate the ap- with camera views indicated by red triangles. plicability of the dataset for MOT tasks by benchmarking a base- line neural learning method. The dataset is publicly available at https:// github.com/ cbelez/ AirTrackSim25. Proceedings of Middle-European Conference on Applied Theoretical Com- puter Science (MATCOS-25). ACM, New York, NY, USA, 4 pages. https: CCS CONCEPTS //doi.org/XXXXXXX.XXXXXXX • Computing methodologies → Computer vision; • Informa- tion systems → Sensor networks. 1 INTRODUCTION Automated wide-area visual surveillance systems have become in- KEYWORDS creasingly important for public and private infrastructures, as they synthetic data generation, visual surveillance, multi-target tracking enable continuous monitoring of large geographic regions through neural learning and computer vision methods. By integrating real- ACM Reference Format: time target detection and tracking with scene interpretation, these Ahmed Mansour, Csaba Beleznai, Fabio F. Oberweger, Verena Widhalm, Nadezda Kirillova, and Horst Possegger. 2025. A Synthetic Multi-View Track- systems facilitate rapid incident response, thus enhancing safety, ing and 3D Pose Dataset for Automated Airport Visual Surveillance. In security, and operational efficiency. Furthermore, their scalability with respect to the number of camera views makes them partic- ularly well suited for managing complex environments, such as Permission to make digital or hard copies of all or part of this work for personal or airports, shopping centers, and transportation hubs. classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation In this work, we address the automated analysis of airport scenes, on the first page. Copyrights for components of this work owned by others than the focusing on the multi-object tracking (MOT) task. This problem author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or setting is confronted with data scarcity and presents several unique republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. research challenges. Although extensive and diverse datasets are MATCOS-25, Oct. 09–10, 2025, Koper, Slovenia available for common object categories such as pedestrians and © 2025 Copyright held by the owner/author(s). Publication rights licensed to ACM. cars [1],[3], publicly accessible airport-specific datasets with trajec-ACM ISBN 978-1-4503-XXXX-X/2018/06 https://doi.org/XXXXXXX.XXXXXXX tory annotations remain limited to the AGVS-T22 dataset [6] and 80 MATCOS-25, Oct. 09–10, 2025, Koper, Slovenia Mansour et al. Figure 3: Top-view of the scene showing specifically posi- tioned path points (orange) and parking points (blue). These Figure 2: Systemic diagram of the Blender-based synthetic points are used to establish randomized directional connec- airport data generator. tions between neighboring nodes, to create realistic airplane movements. A sample generated trajectory is shown in green. the TartanAviation dataset [7]. Furthermore, recent neural learning frameworks, such as FairMOT [11] and CenterTrack [10, 12], formu- the flexibility of the synthetic setup, we are able to generate di- late detection and tracking within a multitask learning paradigm, verse viewpoints, temporal sequences, and annotation-rich data which requires annotation-intensive track-labeled datasets. Lastly, that would be prohibitively costly and logistically challenging to constructing well-balanced training corpora that capture diverse obtain through real-world data collection. In the following, we and challenging scenarios — such as interacting aircrafts at diverse describe the individual aspects of the data generation process, as spatial scales, motion dynamics, and degraded visibility conditions shown in Figure 2: — is difficult to accomplish manually. In light of these challenges, re- Parameters and Domain Randomization: Most scene param- lated work has focused on common object categories, such as in the eters—including lighting conditions, target kinematics, and cam- series of MOTChallenge [1] and the KITTI benchmark [3]. Small ob- era geometries—are randomized within predefined minimum and ject detection and tracking also received attention recently, such as maximum bounds using uniform distributions. This randomization in the SMOT challenge [5], where the limited amount of appearance strategy ensures broad variability in the generated content while cues due to the small target size renders the task difficult. preserving physical plausibility and photorealism. In this work, we present the AirTrackSim25 synthetic airport Scene Environment: The first critical aspect of the environment dataset (see Figure 1), which comprises 14 distinct camera view- is the lighting configuration. Sky- and sun-based illumination is points and a total of 23,278 image frames with 131,866 annotated controlled through Blender’s SkyTexture node, allowing the simula- aircraft instances, capturing realistic airport operational scenarios. tion of different times of day and varying atmospheric conditions The dataset is designed to advance research in small object detec- (e.g., air, dust, and ozone densities). Furthermore, a volumetric fog tion and tracking by providing spatially precise 2D bounding box effect with randomized density is incorporated to emulate realistic (BB) and part-level annotations, occlusion status, 3D BB informa- atmospheric disturbances, thereby reducing target visibility. tion, and consistent target identities with associated trajectories. To Scene Objects and Trajectories: Aircrafts serve as the primary assess the utility of the dataset, we present baseline synthetic neural objects of interest in the scene, as they constitute the targets to be learning experiments for a 2D MOT task, followed by validation detected and tracked. Four 3D airplane models from the Blender on the real-world AGVS-T22 dataset [6]. traffiq add-on [8] are employed, with randomized colors and both The paper is structured as follows: Section 2 provides a brief uniform and slightly non-uniform scaling applied to enhance vari- description of the synthetic generation scheme. Section 3 describes ability. A random number 𝑁 ∈ [10, 20] of aircraft instances are the baseline experiments and their evaluation. Finally, Section 4 introduced into each scene and animated according to the following concludes the paper. principles: The airport environment is structured into three main functional 2 DATA GENERATION SCHEME areas: parking aprons, taxiways, and runways. To generate realistic To construct our synthetic airport dataset, we employ Blender [2] as motion paths, points assigned to one of the matching categories the primary modeling and rendering environment. Within Blender, {𝑝𝑎𝑟 𝑘𝑖𝑛𝑔, 𝑝𝑎𝑡 ℎ, 𝑎𝑖𝑟}, are manually placed across the functional ar- we integrate a commercially available, metrically accurate, and eas of the airport (see Figure 3), defining control points for plausi- highly detailed 3D model of the London Heathrow Airport [4], ble trajectories from parking positions to take-off locations,or vice which serves as the foundation for creating a populated and dy- versa. Together, the control points form a structured graph that en- namic scene. The entire simulation is based on Blender’s python codes the airport’s functional topology. For each aircraft, a random backend. Our developed framework enables the controlled simula- sub-graph connecting a chosen parking position with an air-labeled tion of realistic aircraft operations, including variations in spatial node is selected, to construct a smooth spline-representation as a scale, motion patterns, and interaction scenarios. By leveraging trajectory along which the airplane is animated. The air-labeled 81 A Synthetic Multi-View Tracking and 3D Pose Dataset for Automated Airport Visual Surveillance MATCOS-25, Oct. 09–10, 2025, Koper, Slovenia Figure 4: Example synthetic frames with overlaid ground- truth objects and trajectory annotations. nodes are positioned above ground level to simulate take-off and landing, ensuring that generated trajectories include realistic lift-off and descent dynamics. Trajectory generation follows a procedural scheme: a random parking-labeled node is first selected, and its k (k=5) nearest path-labeled neighbors are evaluated to construct an initial trajectory segment. Since nearest-neighbor selection alone Figure 5: Bounding box height and width distributions within does not enforce the airport’s functional constraints, certain edges the training (top) and test (bottom) datasets. Each plot shows with invalid spatial or directional dependencies are explicitly dis- two semi-transparent histograms, to reveal all distribution allowed. This prevents unrealistic movements, such as an aircraft details. leaving an apron or taxiway at an implausible location or direc- tion. At each step, a permissible next node is chosen from the # views # frames img. resolution # objects 2D BB 3D BB 2D key-points occlusion stat. track-ID k 14 23,278 1088 × 608 px 131,866 ✓ ✓ ✓ ✓ ✓ candidates, and the process is repeated until an air-labeled node Table 1: The main AirTrackSim25 dataset attributes. is reached, marking the completion of a take-off trajectory. Con- versely, landing trajectories are generated in reverse: starting from image frames from the AirTrackSim25 dataset, holding out an inde- an air-labeled node at a random time, the sub-graph is incrementally pendent validation set of 2,428 images. We used an input resolution extended toward a parking-labeled node, resulting in a plausible −4 of 1088 × 608 pixel, the dla34 backbone, initial learning rate of 10 landing sequence. and a batch size of 14. We refer to this model as V50. For compari- Object kinematics play a crucial role in the simulation: since the son, an otherwise identical CenterTrack model was trained with scene is based on a metrically scaled environment, both metric ve- the same settings on the real-world MAV dataset [9] to establish a locities and velocity limits are utilized to generate diverse dynamic real-domain detector baseline; we denote this model as CT. behaviors in the aircraft’s motion. Consequently, the edges of the Test dataset: For evaluation, we used eight scenarios from the defined sub-graph also carry manually-assigned speed informa- AGVS-T22 dataset [6], comprising a total of 10,642 manually anno- tion, allowing for realistic motion patterns, such as slow movement tated image frames with bounding boxes and track IDs, resized to along the taxiway followed by a rapid take-off. the image resolution of 1088×608 pixel. Scene Observers: Fourteen virtual cameras were strategically posi- Comparison of training and test data: To assess the representa- tioned at various airport locations to simulate realistic surveillance tional quality of airplane objects in the proposed synthetic dataset, setups, capturing aircraft with typical size distributions. To ensure we computed distributional statistics of bounding box dimensions significant view diversity and prevent the neural learning model across all frames. Using the same input resolution, equivalent sta- from overfitting to specific camera angles, the view for each gener- tistics were also obtained for the test dataset. The resulting distri- ated sequence was varied by randomizing both camera intrinsics butions for training and test data are shown in Figure 5. As it can (focal length) and extrinsics (orientation) within predefined bounds. be seen, the proposed AirTrackSim25 dataset predominantly con- Figure 4 illustrates representative synthetic views together with tains small objects, with a median bounding box height of 14 pixels, their corresponding ground-truth overlays. whereas the test dataset features objects approximately three times larger (median height: 34 pixels). This highlights the suitability of 3 BASELINE EXPERIMENT AND RESULTS the AirTrackSim25 dataset for advancing research on small object The attributes of the data produced by the synthetic generation detection and tracking. Furthermore, these findings motivate future pipeline are shown in Table 1. To assess the representational quality extensions of the dataset to incorporate larger aircraft instances. and the minimal sim-to-real gap of the generated image data and MOT results: Figure 6 and Table 2 present a qualitative and quan- its annotations, we conducted a neural learning experiment using titative comparison between the CT and V50 models, trained on the CenterTrack [12] unified detection and tracking framework. real and synthetic data, respectively. As illustrated, the syntheti- Model training: We trained a CenterTrack model [12] on 20,850 cally trained V50 model exhibits substantially higher sensitivity, 82 MATCOS-25, Oct. 09–10, 2025, Koper, Slovenia Mansour et al. SEQ/METHOD FRAMES idf1 idp idr RECALL PREC OBJECTS mostly_tracked part_tracked mostly_lost N_falsepos N_misses N_switches N_fragm MOTP baseline-8/CT 2751 0.65 0.61 0.71 0.88 0.76 31115 9 3 0 8683 3582 13 33 0.35 baseline-8/V50 2751 0.46 0.33 0.78 0.98 0.41 31115 12 0 0 43108 602 33 41 0.41 multi-scale-1/CT 1107 0.52 0.57 0.47 0.65 0.79 20874 12 6 4 3622 7218 16 47 0.39 multi-scale-1/V50 1107 0.44 0.37 0.54 0.77 0.52 20874 14 6 2 14599 4868 68 118 0.45 multi-scale-2/CT 1685 0.58 0.69 0.51 0.58 0.78 6156 3 1 1 1018 2589 7 3 0.19 multi-scale-2/V50 1685 0.30 0.19 0.71 0.82 0.23 6156 3 1 1 17295 1081 37 14 0.34 weather-1/CT 1516 0.34 0.41 0.28 0.55 0.79 12208 2 5 2 1760 5541 11 9 0.30 weather-1/V50 1516 0.28 0.22 0.41 0.97 0.51 12208 9 0 0 11185 347 44 35 0.53 weather-3/CT 1217 0.74 0.92 0.61 0.64 0.97 3430 2 0 1 75 1218 4 0 0.17 weather-3/V50 1217 0.43 0.30 0.77 0.86 0.33 3430 2 1 0 5881 496 5 4 0.50 motion-6/CT 461 0.73 0.99 0.54 0.47 0.98 3175 2 2 4 29 1698 2 232 0.26 motion-6/V50 461 0.68 0.77 0.56 0.45 0.69 3175 2 2 4 649 1743 5 238 0.42 ptz-2/CT 1161 0.29 0.23 0.39 0.49 0.29 5553 2 1 4 6603 2814 1 3 0.28 ptz-2/V50 1161 0.15 0.11 0.25 0.40 0.17 5553 1 2 4 10696 3312 14 43 0.43 lc-5/CT 744 0.07 0.04 0.21 0.32 0.06 682 0 1 0 3334 466 1 2 0.22 lc-5/V50 744 0.01 0.01 0.15 0.29 0.01 682 0 1 0 14920 484 3 29 0.63 Table 2: MOT Metrics Summary at IoU=0.75 resulting in improved recall. However, this comes at the cost of re- duced precision, as false alarms increase. In particular, when object size decreases, the model shows reduced specificity for airplanes, occasionally detecting and tracking other small objects (e.g., birds or ground vehicles). Additional false alarms are also triggered by object-like image structures and noise artifacts. Conversely, the CT model trained on real data maintains a low false alarm rate but fails to reliably detect distant or partially occluded targets. The MOT evaluation across eight scenarios, summarized in Table 2, confirms these trends. The increased sensitivity of the V50 model leads to a higher number of mostly-tracked targets—including distant and partially occluded ones—while also producing a higher rate of sta- tionary false alarms. Challenging scenarios such as motion-6, ptz-2, and Figure 6: Detection/tracking results by two baselines. lc-5 , which contain motion or imaging artifacts absent in syn- thetic training data, are particularly problematic for the V50 model. Future work will therefore concentrate on extending the dataset [2] Stichting Blender Foundation. 2025. Blender - a 3D modelling and rendering to achieve a more balanced distribution of aircraft sizes and to in- package. https://www.blender.org. Accessed: 2025-08-21. corporate representative real-world artifacts, with the objective of [3] Andreas Geiger, Philip Lenz, and Raquel Urtasun. 2012. Are we ready for Au- supporting models that attain both high recall and high precision. puter Vision and Pattern Recognition (CVPR) tonomous Driving? The KITTI Vision Benchmark Suite. In Conference on Com-. [4] META Group. 2020. London Heathrow Airport - LHR 3D model. https://www. 4 turbosquid.com/3d-models/london-heathrow-airport-lhr-1543630. Accessed: CONCLUSIONS 2025-08-21. We introduce a synthesis framework and the AirTrackSim25 syn- [5] Riku Kanayama, Yuki Yoshida, and Yuki Kondo. 2025. Baseline code for SMOT4SB thetic dataset, designed to address the scarcity of annotated data by IIM-TTIJ. https://www.mva-org.jp/mva2025/challenge [6] Tingyu Li, Xiang Zhang, Zihao Tang, and Yudie Liu. 2023. AGVS-T22: A New for airport surveillance and to support research in object detection, Multiple Object Tracking Dataset for Airport Ground Video Surveillance. In 2023 multi-object tracking, and data association. The dataset offers di- IEEE 26th International Conference on Intelligent Transportation Systems (ITSC). verse wide-area airport perspectives with realistic aircraft dynamics 4380–4387. https://doi.org/10.1109/ITSC57777.2023.10421871 ISSN: 2153-0017. [7] Jay Patrikar, Joao Dantas, Brady Moon, Milad Hamidi, Sourish Ghosh, Nikhil and comprehensive annotations, including 2D/3D bounding boxes, Keetha, Ian Higgins, Atharva Chandak, Takashi Yoneyama, and Sebastian Scherer. key-points, and trajectory information. Its applicability to MOT 2025. Image, speech, and ADS-B trajectory datasets for terminal airspace opera- tasks is demonstrated through benchmarking with a baseline neu- [8] polygoniq xyz s.r.o. 2025. traffiq: vehicle library for blender. https://polygoniq. tions. Scientific Data 12, 1 (2025), 468. ral learning framework, underscoring its relevance for advancing com/software/traffiq. Accessed: 2025-08-21. small object detection and tracking. [9] Daniel Steininger, Verena Widhalm, Julia Simon, Andreas Kriegler, and Christoph Sulzbachner. 2021. The Aircraft Context Dataset: Understanding and Optimizing Data Variability in Aerial Domains. In Proceedings of the IEEE/CVF International ACKNOWLEDGMENTS Conference on Computer Vision (ICCV) Workshops. 3823–3832. [10] Yihong Xu, Yutong Ban, Guillaume Delorme, Chuang Gan, Daniela Rus, and This work was supported by the SAFER project, funded by the Xavier Alameda-Pineda. 2022. TransCenter: Transformers with Dense Represen- Austrian Research Promotion Agency (FFG) under project number tations for Multiple-Object Tracking. https://doi.org/10.48550/arXiv.2103.15145 4452868. arXiv:2103.15145 [cs]. [11] Yifu Zhang, Chunyu Wang, Xinggang Wang, Wenjun Zeng, and Wenyu Liu. 2021. FairMOT: On the Fairness of Detection and Re-Identification in Multiple Object REFERENCES Tracking. International Journal of Computer Vision 129, 11 (Nov. 2021), 3069–3087. https://doi.org/10.1007/s11263-021-01513-4 arXiv:2004.01888 [cs]. [1] P. Dendorfer, H. Rezatofighi, A. Milan, J. Shi, D. Cremers, I. Reid, S. Roth, K. [12] Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. 2020. Tracking Objects Schindler, and L. Leal-Taixé. 2020. MOT20: A benchmark for multi object tracking as Points. https://doi.org/10.48550/arXiv.2004.01177 arXiv:2004.01177 [cs]. in crowded scenes. arXiv:2003.09003[cs] (March 2020). http://arxiv.org/abs/1906. 04567 83 Sphere Target-Based Point Cloud Registration in a Railway Safety Application David Podgorelec Luka Lukač Sašo Pečnik david.podgorelec@um.si luka.lukac@um.si saso.pecnik@um.si University of Maribor University of Maribor University of Maribor Faculty of Electrical Engineering Faculty of Electrical Engineering Faculty of Electrical Engineering and Computer Science and Computer Science and Computer Science Maribor, Slovenia Maribor, Slovenia Maribor, Slovenia Blaž Repnik Borut Žalik blaz.repnik@um.si borut.zalik@um.si University of Maribor University of Maribor Faculty of Electrical Engineering Faculty of Electrical Engineering and Computer Science and Computer Science Maribor, Slovenia Maribor, Slovenia Figure 1: A part of the test environment with three (of six) sphere targets and a LiDAR scanner (of two) left behind. Abstract 1 Introduction The paper introduces a variation of registration of two LiDAR Similar to the human visual system, spatial data capture devices point clouds based on sphere targets, which uses our own geo- such as cameras and scanners acquire information within a lim- metric construction method to determine the centres and radii of ited field of view (FoV). Furthermore, the density, and hence the the spheres. Tests gave encouraging results for use in a railway quality and usefulness, of the acquired data decreases with dis- safety application, as the registration error is below 30 % of the tance from the capturing sensor. Therefore, combining individual voxel size used there. Unlike the traditional landmark identifica- shots of the same area in order to improve the processing and tion approach based on solving the system of linear equations, analysis of spatial data has been common practice since the pre- the proposed approach offers good geometric interpretability computer era. Examples include image stitching, e.g., in creating and error explainability, which has potential for the development panoramic images, the use of overlapping transparency layers in of heuristics that would prune the solution space and, eventu- cartography and film effects, as well as stereographic and pho- ally, enable us to use the time saved to conduct a more detailed togrammetric techniques. With the development of 3D geometric investigation in the vicinity of the current optima. modelling and the advent of 3D data capture devices, a conve- nient capability has emerged to capture a 3D scene from multiple Keywords viewpoints and merge the captured data. This is achieved by spatially aligning the local coordinate systems of two or more point cloud registration, sphere target, LiDAR, sphere centre geometric models with respect to a reference coordinate sys- determination, radius determination, level crossing tem and thereby aligning the models themselves, which is called geometric data registration or simply registration [11]. Permission to make digital or hard copies of all or part of this work for personal In this work, we focus on geometric models in the form of or classroom use is granted without fee provided that copies are not made or point clouds. Furthermore, we focus on rigid transformations, distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this which are common when dealing with time-aligned point clouds work must be honored. For all other uses, contact the owner /author(s). [2]. For simplicity, we consider the registration of two point Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia clouds, as additional ones can be handled incrementally. Specifi- © 2025 Copyright held by the owner/author(s). cally, we want to align (register) two point clouds captured by 84 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Podgorelec et al. a pair of LiDAR (Light Detection and Ranging) scanners that cloud belong to individual target spheres. Our focus is primarily monitor events at a level crossing between a railway and a road. on accuracy rather than speed, automation, and generality of The problem here is that the central area is quite far from both the process. Thereby, we used specific target geometry, such as scanners, which means that the points captured there are quite spheres’ instalation heights and their radii. Unwanted points of far apart, causing problems in detecting unwanted obstacles in the stand are above or below the sphere and in the area around each single scan. By registering and merging two point clouds, its central vertical axis. As a last resort, there is also the option of we obtain a denser merged point cloud that improves detection. interactively deleting some unwanted points. More advanced ap- Comprehensive reviews of point cloud registration methods proaches, based on similar principles as registration with natural can be found, e.g., in [9, 5, 3, 4]. They are usually divided into targets can be found in, e.g., [7]. target-based and target-free methods [8]. Targets are (easily) identifiable markers artificially inserted into the scene to simplify 2.3 Landmark Identification alignment. The target-based methods align pairs of models of the Although two LiDARs capture different points on the surface of same target from both point clouds, indirectly aligning the entire a single sphere target, the centre of the sphere is unique. In this scene. The target-free methods are based on features that are Subsection, we present a procedure for determining the sphere referred to as natural targets by some authors. These include, e.g., centre from four non-coplanar points on its surface. Three non- edges, prominent vertices, and contrasting areas, which are more collinear sphere centres, given in the local coordinate systems of difficult to detect because their geometry, size, and location are both LiDARs, can be used as landmarks to establish an interme- less known. The target-free methods are typically more general, diate coordinate system. Based on this, we will then introduce also useful in dynamic environments and, recently, increasingly the registration transformation matrix 𝑀 in Subsection 2.4. based on deep learning [10]. In this paper, we focus on target- Let 𝑃 ( ) ( ) ( ) ( ) 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 𝑥 , 𝑦 , 𝑧 , 𝑃 𝑥 , 𝑦 , 𝑧 , 𝑃 𝑥 , 𝑦 , 𝑧 , and 𝑃 𝑥 , 𝑦 , 𝑧 based methods. Using the points detected by both devices on the 3 represent four non-coplanar points in . We want to describe the R target surface, they can calculate additional points (landmarks) ( ) sphere 𝑆 𝑟 , 𝐶 through them, which means we want to determine where there is a perfect match, and then align the scene accurately ( ) its centre 𝐶 𝑥𝐶, 𝑦𝐶, 𝑧𝐶 and radius 𝑟. Applying a well-known equa- by aligning these landmarks. tion of a sphere to such quartet of points gives the system of four In Section 2, we present the registration method we developed quadratic equations (1). Subtracting the other three equations and used, which consists of preprocessing, sphere target extrac- separately from the first one gives a system of linear equations tion, landmark identification, and registration itself. In Section 3, (2) in three unknowns 𝑥 , 𝑦 , and 𝑧 . 𝐶 𝐶 𝐶 we describe the test environment, present the registration results, and analyze their accuracy. In Section 4, we summarize the work ( 2 2 2 2 − ) + ( − ) + ( − ) = = 𝑥 𝑥 𝑦 𝑦 𝑧 𝑧 𝑟 , 𝑖 1, ..., 4 (1) 𝑖 𝐶 𝑖 𝐶 𝑖 𝐶 and present challenges for further improvements and research. The main scientific contributions of the article are the original geometric construction procedure for calculating the centre and 2 𝑥 1 𝑥2 , 2 𝑦1 𝑦2 , 2 𝑧1 𝑧2  ( − ) ( − ) ( − ) 𝑥𝐶 radius of a sphere and the analysis of accuracy.     ( − ) ( − ) ( − ) 2 𝑥1 𝑥3 , 2 𝑦1 𝑦3 , 2 𝑧1 𝑧3  𝑦𝐶     ( − ) ( − ) ( − ) 2 𝑥1 𝑥4 , 2 𝑦1 𝑦4 , 2 𝑧1 𝑧4  𝑧𝐶 2 Methodology     2 2 2 2 2 2  𝑥1 𝑦1 𝑧1 𝑥2 𝑦2 𝑧2  ( + + ) − ( + + ) We use two stationary LiDARs, which means we can perform   2 2 2 2 2 2 = ( + + ) − ( + + )  𝑥1 𝑦1 𝑧1 𝑥3 𝑦3 𝑧3  (2) registration before launching the application itself. For accuracy,  2 2 2 2 2  2 ( + + ) − ( + + )  𝑥1 𝑦1 𝑧1 𝑥4 4 𝑦 𝑧4  we use artificial targets, as we have enough time to safely set   The system (2) can then be solved using traditional methods, them up, identify them accurately, and then remove them. Based such as Gaussian elimination, inverse matrix calculation, etc. on the literature [1, 8], in which various target types and layouts Once we have the centre 𝐶 of the sphere, we calculate the radius were assessed, and our own experiments, in which we focused 𝑟 by inserting the coordinates of any point 𝑃1, 𝑃2, 𝑃3, or 𝑃4 into on the performance of both LiDARs used and the possibility of the corresponding line of (1). separating target and stand points in a simulated railway level This procedure is easy to understand and not difficult to imple- crossing layout, we chose sphere targets. The procedure is carried ment, but it also has its drawbacks. The results in Subsection 3.2 out in four phases described in Subsections 2.1–2.4. show that selecting different four-point sets on the sphere pro- 2.1 duces significantly different results. Thus, the registration is an Preprocessing optimization problem. Below, we present our own construction This step depends on the initial state of the data. It may include approach, which has a clear geometric interpretation. Errors are mapping the data from perspective to orthographic projection easier to explain, as individual points of a quartet have different, in order to establish Cartesian coordinate systems 𝐶𝑆 for each 𝑖 precisely defined roles. device (𝑖 2), scaling to unify the unit of measurement in all 𝐶𝑆 , ≥ 𝑖 Athough our alternative method is completely intuitive, we cropping, and various filtering, e.g., denoising. In our case, each have not yet met it in literature. It first considers three non- 𝐶𝑆 is centred at the corresponding LiDAR lens and is left-handed. 𝑖 collinear points 𝑃1, 𝑃2, 𝑃3 from a quartet on the sphere target The 𝑍-axis points towards the centre of the FoV (diagonally ( ) Σ 𝑆 𝑟 , 𝐶 . They define a plane , which divides 𝑆 into two parts: 123 downward), the 𝑋-axis horizontally to the left, and the 𝑌-axis a larger one forming a spherical zone of one base, within which in the direction of 𝑍 𝑋 , i.e., diagonally upward. The unit of × the sphere centre 𝐶 is located, and a smaller one forming a measurement is metre. ∩ Σ spherical cap. The intersection 𝑆 123 is the circumreference 𝑐 123 𝑃1, 𝑃2, 𝑃3 𝑟123, 𝐶123 of the triangle 𝑃1𝑃2𝑃3. = circ ( ) = circ ( ) Δ 2.2 Target Extraction 𝐶 and 𝑟 represent the centre and radius of 𝑐 , respectively. 123 123 123 The next step is to isolate individual target geometric models Figure 2a shows (initially) known geometric elements and at- from the scene. In our case, we determine which points in a point tributes in black, those calculated so far in blue, and those not 85 Sphere Target-Based Point Cloud Registration Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia yet determined in red. These three colours are used consistently Centre and radius of the sphere from 4 points Algorithm 1 in Figures 2b–c. An important observation at this stage is that function SphereThroughFourPoints( 𝑃 1, 𝑃2, 𝑃3, 𝑃4) the centre 𝐶 of the sphere must represent the vertex of a right Δ 𝐶 123 centre of circumreference 𝑐123 of 𝑃1𝑃2𝑃3 ← circular cone above 𝑐 123, i.e. it lies on 𝑙123. In the second step, we 𝑟 123 𝐶123𝑃1 ⊲ Radius of 𝑐123 ← | | determine the orthogonal distance 𝑓 from the fourth point 𝑃 of the quartet to 123, the distance 𝑒 𝑃4𝐶123 , and angle 𝛽 (Fig- 123 n = Σ 4 ( − ) × ( − ) | | ← ⊲ Norm. vect. Δ 𝑃1 𝐶123 𝑃2 𝐶123 𝑃 𝑃 𝑃 1 2 3 | (𝑃1 − 𝐶123) × (𝑃2 −𝐶123) | ure 2b). Finally, we use 𝑒, 𝛽 , and the cosine theorem in triangles Δ if |𝐶123𝑃4| = 𝑟123 then Δ 𝑃4𝐶123𝐶 and 𝑃1𝐶𝐶123 to determine the distance 𝑑 𝐶𝐶123 = || and the sphere target radius return (𝐶123, 𝑟123) 𝑟 . Through 𝑑 , we then also determine the sphere centre 𝐶 (Figure 2c). end if 𝑇 orthogonal projection of 𝑃 on the plane of 𝑃 𝑃 𝑃 ← Δ 4 4 1 2 3 𝑒 𝐶 123𝑃4 ← || 𝑒 𝑟 123 − 2 2 𝑑 ← 2 𝑃4𝑇4 | | √︁ 𝑟 𝑑 𝑟 + ← 2 2 123 if (n123 · (𝑃4 − 𝐶123) > 0) ⊕ (𝑒 > 𝑟123) then 𝐶 𝐶 123 𝑑 123 ← + n else 𝐶 ← 𝐶123 − 𝑑 n123 end if return (𝐶, 𝑟 ) end function (see Equation 3 and Figure 3) is used to establish 𝐶𝑆 with the 𝐼 origin 𝑂 and orthogonal unit coordinate vectors 𝑈 , 𝑉 and 𝑊 . 𝑀 is then computed as a composition of two transformations – 𝑀 2 𝑆𝐼 from 𝐶𝑆 to 𝐶𝑆 , and 𝑀 from the latter to 𝐶𝑆 . Note that the 𝑆 𝐼 𝐼 2𝑅 𝑅 system of formulas (3) and the interpretation from Figure 3 must be employed separately for 𝐴 , 𝐵 , 𝐶 , 𝑂 , 𝑈 , 𝑉 , 𝑊 expressed { } 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 in 𝐶𝑆𝑆 , and 𝐴𝑅, 𝐵𝑅, 𝐶𝑅, 𝑂𝑅, 𝑈𝑅, 𝑉𝑅, 𝑊𝑅 expressed in 𝐶𝑆𝑅. { } Figure 3: Construction of the intermediate coordinate sys- Figure 2: Determination of the sphere centre 𝐶 and radius tem from non-collinear 𝐴, 𝐵 and 𝐶 (adapted from [6]). 𝑟 𝑃1 𝑃2 𝑃3 𝑃4 𝑃4 from points , , , and on the sphere, when and 𝐶 123 123 are on the same side of Σ : a) determining Σ, b) de- scribing the relation between 𝑃4 and Σ123, c) final solution. 𝑂 = 𝐴, 𝐵 𝑂 − 𝑈 , = Algorithm 1 explains this procedure in a compact form. Note | − | 𝐵 𝑂 that Figure 2 only shows the case where 𝑃4 and 𝐶 are on the same (3) side of 123. The situation with 𝑃4 in the spherical cap is handled − ) Σ 𝑈 𝐶 𝑂 × ( 𝑊 , = by assigning the absolute value to 𝑑, while the exception with 𝑈 𝐶 𝑂 | × ( −) | both 𝑃4 and 𝐶 in 123 is caught with the first if-clause. 𝑉 =𝑊 × Σ 𝑈 . The transformation 𝑀 from 𝐶𝑆 to 𝐶𝑆 is given in homoge- 2.4 𝐼2𝑅 𝐼 𝑅 Registration neous coordinates as the composition of a 3D rotation 𝑅𝑜𝑡 𝑅 and The determination of the registration matrix 𝑀 is adopted from translation 𝑇 𝑟 𝑎𝑛 𝑂 , as shown in (4). The matrix 𝑀 from ( ) 𝑅 𝑅 𝐼 2𝑆 [6]. There, it was used to map the point cloud from the 𝐶𝑆 𝑆 co- 𝐶𝑆𝐼 to 𝐶𝑆𝑆 can be generated in the same manner, but the inverse ordinate system of the LiDAR scanner to the robot’s 𝐶𝑆 via 𝑀 𝑅 2 𝑆𝐼 , as shown in (5), is actually needed. 𝑀 is then obtained as the intermediate 𝐶𝑆 𝐼 . Here, 𝑆 (source) denotes the first LiDAR, 𝑅 the composition 𝑀 𝑀 𝑀 . = 𝐼2𝑅 𝑆2𝐼 (reference, registered) the target coordinate system of the second LiDAR, and 𝐼 the intermediate coordinate system. 𝑀 is deter- 𝑈𝑅.𝑥 𝑉𝑅.𝑥 𝑊𝑅.𝑥 𝑂𝑅 .𝑥   mined by three translations and three rotations along/around   𝑈𝑅.𝑦 𝑉𝑅 .𝑦 𝑊𝑅.𝑦 𝑂𝑅 .𝑦 the coordinate axes. The role of registration is thus to determine  𝑀 = 𝐼 2 𝑅 𝑇 𝑟 𝑎𝑛 𝑅 ( 𝑂 𝑅 ) · 𝑅𝑜𝑡 𝑅 =  (4)   𝑈 .𝑧 𝑉 .𝑧 𝑊 .𝑧 𝑂 .𝑧 the parameters of these six elementary transformations with the  𝑅 𝑅 𝑅 𝑅    best possible accuracy. A triplet of landmark points 𝐴, 𝐵, and 𝐶  0 0 0 1    86 Information Society 2025, 6–10 October 2025, Ljubljana, Slovenia Podgorelec et al. Table 1: Sphere identifier, calculated radius [cm], and the −1 −1 −1 𝑇 distance between both calculated centres [cm] 𝑀 2 𝑆𝐼 𝑀 𝑅𝑜𝑡 𝑇 𝑟 𝑎𝑛 𝑂 𝑅𝑜𝑡 2 = 𝐼 𝑆 = 𝑆 · 𝑆 ( 𝑆 ) = 𝑇 𝑟 𝑎𝑛 𝑂 𝑆 · 𝑆 (−𝑆) 𝑈 𝑆 .𝑥 𝑈𝑆.𝑦 𝑈𝑆 .𝑧 0 1 0 0 𝑂𝑆 .𝑥 Sphere ID Calculated radius Centre offset −     𝑉𝑆 .𝑥 𝑉𝑆 .𝑦 𝑉𝑆 .𝑧 0 0 1 0 𝑂𝑆 .𝑦 1 −     (5) 25.02 1.15 =   ·       2 25.02 1.47 𝑊 .𝑥 𝑊 .𝑧 0 0 0 1 𝑂  𝑆 𝑊 𝑆 .𝑦 𝑆   − 𝑆 .𝑧      3 25.03 0.34  0 0 0 1 0 0 0 1      4 25.04 3.13 3 Results 5 24.99 1.73 6 25.02 2.30 3.1 Test Setup In tests conducted on the lawn of the primary school in Kamnica results for use in a railway safety application for level crossing (Figure 1), we simulated conditions at an actual railway level monitoring. The registration error is below 30 % of the voxel side crossing, which would be monitored by two LiDARs at the di- length used there. Unlike the traditional landmark identification agonal ends of the crossing. The optical properties, installation approach based on solving the system of linear equations, the height, and viewing angle of LiDARs were also adapted to the proposed approach offers good geometric interpretability and expected conditions. We used two Ouster OS1 LiDARs with a error explainability, which has potential for the development of vertical resolution of 128 channels, a horizontal resolution of ◦ ◦ heuristics that would prune the solution space and, eventually, 2048 samples, and a field of view of 42.4 vertically and 360 enable us to use the time saved to conduct a more detailed inves- horizontally. They were 40 m apart, at a height of 3.2 m, and tigation in the vicinity of the current optima. Current experience inclined to cover the ground from 3 m onwards. Approximately suggests that the best quartets are those where all normals are di- halfway between them, we placed six styrofoam sphere targets rected as closely as possible toward the LiDAR, while at the same with radii of 25 cm and at different heights, making sure that time the points are not too close together and are as non-coplanar no three sphere centres were collinear. We manually measured as possible (defining a tetrahedron with a larger volume). their circumferences in several directions and concluded that the deviations of radii were below 2 mm. Given the optical properties of LiDARs, the nearest neighbour points captured at a distance of Acknowledgements 20 m are approximately 6 cm apart in the horizontal direction and The research was funded by the Slovene Research and Innovation 11 cm in the vertical direction. In our railway safety application, Agency under Research Project J2-4458 and Research Programme we use voxels with sides of 12 cm, which are sufficient for reliable P2-0041. The authors are grateful to Fokus Tech d.o.o. from Celje detection of standardized minimum obstacles of 100 50 50 cm. and OŠ Kamnica for providing equipment and testing facilities. × × The spheres were mounted on slender stands, which we "erased" using the procedure described in Subsection 2.2. References [1] Burcin Becerik-Gerber, Farrokh Jazizadeh, Geoffrey Kavulya, and Gulben 3.2 Accuracy Analysis Calis. 2011. Assessment of target types and layouts in 3d laser scanning for Autom. Constr. registration accuracy. , 20, 5, 649–658. doi:10.1016/j.autcon.2 Each LiDAR acquired between 10 and 50 points on each sphere. 010.12.008. [2] Ben Bellekens, Vincent Spruyt, Rafael Berkvens, Rudi Penne, and Maarten We considered 10 frames for each LiDAR, i.e., 10 pairs of point Weyn. 2015. A benchmark survey of rigid 3d point cloud registration algo- clouds. For each pair, we tested all possible quartets of points rithms. Int. J. Adv. Intell. Syst, 8, 5, 118–127. http://72.52.166.99/articles/ints ys_v8_n12_2015_10.pdf . on each sphere and selected the best result. After the Landmark [3] Menthy Denayer, Joris De Winter, Evandro Bernardes, Bram Vanderborght, identification step, we first checked whether both calculated radii and Tom Verstraten. 2024. Comparison of point cloud registration techniques were in the range [24.8 cm, 25.2 cm], and then selected the pair on scanned physical objects. , 24, 7, 2142. doi:10.3390/s24072142. Sensors [4] Mengjin Lyu, Jie Yang, Zhiquan Qi, Ruijie Xu, and Jiabin Liu. 2024. Rigid with the smallest centre offset. The centre offset is the deviation pairwise 3d point cloud registration: a survey. , 110408. Pattern Recognit. between the calculated centres of the same sphere, one originally doi:10.1016/j.patcog.2024.110408. from 𝐶𝑆 [5] Jaroslav Marek and Pavel Chmelař. 2023. Survey of point cloud registration and the other one transformed (registered) from 𝐶𝑆 . 𝑅 𝑆 methods and new statistical approach. , 11, 16, 3564. doi:10.339 Mathematics The Excel spreadsheets of results contain a total of almost 200 0/math11163564. MB of data. Table 1 shows the results of the best alignment. Of [6] David Podgorelec, Suzana Uran, Andrej Nerat, Božidar Bratina, Sašo Pečnik, the two calculated radii, we write down the worse one, i.e. the Marjan Dimec, Franc Žaberl, Borut Žalik, and Riko Šafarič. 2023. Lidar-based maintenance of a safe distance between a human and a robot arm. , Sensors one that deviates more from the expected 25 cm. Both metrics 23, 9, 4305. doi:10.3390/s23094305. gave encouraging results with the radius error below 0.3 mm [7] Trung-Thien Tran, Van-Toan Cao, and Denis Laurendeau. 2016. Esphere: extracting spheres from unorganized point clouds: how to extract multiple and the centre offset below 3.5 cm, which is half better than spheres accurately and simultaneously. , 32, 1205–1222. doi:10 Vis. Comput. LiDAR resolution applied 20 m from the lens, and below 30 % .1007/s00371- 015- 1157- 0. of the used voxel size. Note that the worst among best offsets [8] Tilen Urbančič, Žiga Roškar, Mojca Kosmatin Fras, and Dejan Grigillo. 2019. New target for accurate terrestrial laser scanning and unmanned aerial in individual frames was 6.1 cm (probably due noise). Besides, vehicle point cloud registration. , 19, 14, 3179. doi:10.3390/s19143179. Sensors there were always quartets of points found that gave completely [9] Dongfang Xie, Wei Zhu, Fengxiang Rong, Xu Xia, and Huiliang Shang. 2021. unusable results, as the calculated radii ranged from 5 cm to 2 m. Registration of point clouds: a survey. In 2021 Int. Conf. on Networking Systems of AI (INSAI). IEEE, 136–142. doi:10.1109/INSAI54028.2021.00034. [10] Ningli Xu, Rongjun Qin, and Shuang Song. 2023. Point cloud registration 4 Conclusion for lidar and photogrammetric data: a critical synthesis and performance analysis on classic and deep learning algorithms. ISPRS J. Photogramm. We presented a variation of the registration of two LiDAR point Remote Sens. , 8, 100032. doi:10.1016/j.ophoto.2023.100032. clouds based on sphere targets, which uses our own geometric [11] Dongho Yun, Sunghan Kim, Heeyoung Heo, and Kwang Hee Ko. 2015. Automated registration of multi-view point clouds using sphere targets. construction method to determine the centres and radii of the Adv. Eng. Inform., 29, 4, 930–939. doi:0.1016/j.aei.2015.09.008. spheres. Tests have shown that the method achieves encouraging 87 88 Indeks avtorjev / Author index Alland Lucas ................................................................................................................................................................................ 27 Árgilán Viktor .............................................................................................................................................................................. 15 Baldouski Daniil ............................................................................................................................................................................. 7 Békési József ................................................................................................................................................................................ 15 Beleznai Csaba ............................................................................................................................................................................. 80 Berend Gábor ............................................................................................................................................................................... 44 Bóta András .................................................................................................................................................................................. 48 Brumen Matej ............................................................................................................................................................................... 68 Dabbous Ahmed ........................................................................................................................................................................... 48 Dávid Balázs ................................................................................................................................................................ 7, 60, 64, 75 Dobravec Tomaž .......................................................................................................................................................................... 19 Dömösi Pál ................................................................................................................................................................................... 30 Egri Péter ...................................................................................................................................................................................... 64 Galambos Gábor ........................................................................................................................................................................... 15 Hegyháti Máté ........................................................................................................................................................................ 11, 52 Horvat Štefan ............................................................................................................................................................................... 68 Horváth Géza ............................................................................................................................................................................... 30 Hren Boštjan ................................................................................................................................................................................. 19 Kawa Arkadiusz ........................................................................................................................................................................... 75 Kebelei Csaba ............................................................................................................................................................................... 11 Kirillova Nadezda ........................................................................................................................................................................ 80 Kiss Mihály .................................................................................................................................................................................. 44 Kovačević Nikola ......................................................................................................................................................................... 60 Krész Miklós ...................................................................................................................................................................... 7, 39, 64 Kusper Gábor ............................................................................................................................................................................... 23 Lukač Luka ................................................................................................................................................................................... 84 Mansour Ahmed ........................................................................................................................................................................... 80 Mongus Domen ............................................................................................................................................................................ 68 Nagy Benedek .............................................................................................................................................................................. 35 Oberweger Fabio F. ...................................................................................................................................................................... 80 Papp Imre ..................................................................................................................................................................................... 15 Pečnik Sašo .................................................................................................................................................................................. 84 Podgorelec David ......................................................................................................................................................................... 84 Possegger Horst ............................................................................................................................................................................ 80 Quilliot Alain ............................................................................................................................................................................... 56 Repnik Blaž .................................................................................................................................................................................. 84 Sali Attila ..................................................................................................................................................................................... 27 Strnad Damjan .............................................................................................................................................................................. 68 Szabó Sándor ................................................................................................................................................................................ 72 Szaller Ádám ................................................................................................................................................................................ 64 Tahalea Sylvert Prian ............................................................................................................................................................. 39, 75 Tavzes Črtomir ............................................................................................................................................................................. 60 Toussaint Hélène .......................................................................................................................................................................... 56 Váncza József ............................................................................................................................................................................... 64 Widhalm Verena .......................................................................................................................................................................... 80 Wu Nicole .................................................................................................................................................................................... 27 Žalik Borut ................................................................................................................................................................................... 84 Zaválnij Bogdán ........................................................................................................................................................................... 72 89 Srednjeevropska konferenca o uporabnem teoretičnem računalništvu in informatiki (MATCOS) Middle-European Conference Computer Science Uredniki l Editors: (MATCOS) on Applied Theoretical Andrej Brodnik Gábor Galambos Rok Požar