On matchings and related problems in graphs, hypergraphs, and doubly stochastic matrices Ioannis Panagiotas To cite this version: Ioannis Panagiotas. On matchings and related problems in graphs, hypergraphs, and doubly stochas- tic matrices. Data Structures and Algorithms [cs.DS]. Université de Lyon, 2020. English. NNT : 2020LYSEN068. tel-03011794 HAL Id: tel-03011794 https://tel.archives-ouvertes.fr/tel-03011794 Submitted on 18 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Numéro National de Thèse : 2020LYSEN068 THÈSE de DOCTORAT DE L’UNIVERSITÉ DE LYON opérée par l’École Normale Supérieure de Lyon École Doctorale N◦512 École Doctorale en Informatique et Mathématiques de Lyon Spécialité : Informatique présentée et soutenue publiquement le 09/10/2020, par : Ioannis PANAGIOTAS On matchings and related problems in graphs, hypergraphs, and doubly stochastic matrices Sur les couplages et les problèmes liés dans les graphes, les hypergraphes et les matrices doublement stochastiques Devant le jury composé de : Clémence MAGNIEN Directrice de recherche, CNRS Rapporteure Alex POTHEN Professeur, Purdue Univ., Etats-Unis Rapporteur Marthe BONAMY Chargée de recherche, CNRS Examinatrice Dimitrios THILIKOS Directeur de recherche, CNRS Examinateur Bora UÇAR Chargé de recherche, CNRS Directeur de thèse Fanny DUFOSSE Chargée de recherche, INRIA Co-Encadrante de thèse Contents Acknowledgements......................................v Résumé français....................................... vi 1 Introduction 1 1.1 Undirected graphs...................................1 1.2 Hypergraphs......................................3 1.3 Sparse matrices....................................4 1.3.1 The permanent function...........................4 1.3.2 Doubly stochastic matrices..........................5 1.4 Sparse tensors.....................................6 1.5 Structure of the thesis.................................7 2 Matchings in bipartite graphs9 2.1 A survey on matching heuristics........................... 10 2.2 An examination of the Karp–Sipser algorithm................... 11 2.2.1 An expected O(m log n)-time algorithm................... 12 2.2.2 An implementation with list caching..................... 13 2.2.3 An alternating component approach..................... 14 2.2.4 Fast recovery of the matching......................... 16 2.2.5 Experiments................................... 17 2.2.6 Related work.................................. 25 2.3 Scaling based near-optimal randomized algorithms................. 25 2.3.1 2outMC: Monte Carlo on 2-out graphs................... 26 2.3.2 TruncRW: Truncated random walk with nonuniform sampling...... 31 2.3.3 Experiments................................... 34 2.4 A scaling based derandomized algorithm...................... 43 2.4.1 The derandomization............................. 45 2.4.2 Some preliminary experiments........................ 48 2.5 Concluding remarks.................................. 49 3 Matchings in undirected graphs 51 3.1 One-Out: The main heuristic............................ 51 3.1.1 One-Out: Analysis.............................. 53 3.1.2 Two variants of One-Out ........................... 58 3.2 Experiments...................................... 59 3.2.1 On real-life graphs............................... 59 3.2.2 On a hard synthetic instance for KSR1 .................... 62 3.2.3 On large-scale graphs............................. 63 ii CONTENTS iii 3.3 Concluding remarks.................................. 63 4 Matchings in hypergraphs 65 4.1 Heuristics for maximum d-dimensional matching.................. 66 4.1.1 A Greedy heuristic for Max-d-DM...................... 66 4.1.2 KarpSipserH for Max-d-DM......................... 66 4.1.3 KarpSipserHScaling for Max-d-DM...................... 67 4.1.4 Hypergraph matching via pseudo scaling................... 69 4.1.5 Reduction to bipartite graph matching.................... 69 4.1.6 Performing local search............................ 71 4.2 Experiments...................................... 71 4.2.1 On random hypergraphs............................ 71 4.2.2 On synthetic hypergraphs........................... 74 4.2.3 On real-life hypergraphs............................ 76 4.2.4 Comparison with an independent set solver................. 76 4.3 Concluding remarks.................................. 77 5 Counting the number of perfect matchings in graphs 79 5.1 Theoretical background................................ 80 5.2 Related work...................................... 80 5.3 The proposed algorithm and its analysis....................... 81 5.3.1 The algorithm................................. 82 5.3.2 The analysis................................... 83 5.4 An estimator for undirected graphs.......................... 87 5.4.1 The algorithm and its analysis........................ 87 5.4.2 Filtering out redundant edges......................... 90 5.5 Experiments....................................... 91 5.5.1 On bipartite graphs.............................. 91 5.5.2 On general, undirected graphs......................... 96 5.6 Concluding remarks.................................. 98 6 Results on the Birkhoff–von Neumann decomposition 101 6.1 The two heuristics................................... 102 6.2 Analysis of the two heuristics for computing BvN decompositions........ 102 6.3 Heuristics for GreedyBVN ............................... 106 6.4 Experiments on real-life matrices........................... 107 6.5 Concluding remarks.................................. 109 7 Conclusion 111 7.1 Summary of the chapters................................ 111 7.2 Future work....................................... 112 List of publications 121 List of Algorithms 2.1 2outMC: Monte Carlo on 2-out graphs........................ 27 2.2 Sample: Algorithm to sample a random neighbor of a column vertex c with dc neighbors......................................... 33 2.3 OneSided: Matching heuristic............................. 44 2.4 OneSidedDerand: The derandomized variant of OneSided ........... 47 3.1 One-Out: Heuristic for matching in undirected graphs............... 52 3.2 KarpSipserOne-Out: Specialized Karp–Sipser for 1-out graphs......... 54 4.1 KarpSipserHScaling: The scaling-based extension of Karp–Sipser in hyper- graphs.......................................... 68 5.1 estScalingPerm: Permanent estimation...................... 83 5.2 estScalingMtc: Estimation of the number of perfect matchings in graphs... 88 6.1 Generalized-Birkhoff: Template to find a BvN decomposition........ 102 iv Acknowledgments I would like first and foremost to thank my supervisor Bora Uçar for our collaboration during the last three and a half years. He was always very positive, approachable and supportive to me. He was always more than keen to discuss with me when I was facing some issues. His comments were always helpful and I learned a lot through our conversations about research and writing papers. His enthusiasm for the matching problem is probably unparalleled! I am also thankful for having Fanny Dufossé as my second supervisor. Our frequent visits to each other during the first year, where Bora was away in the USA, really helped me stay focused. I was also very lucky to work with Kamer Kaya from Sabanci University in Turkey who essentially became my unofficial third supervisor. I was really amazed by Kamer’s ability to tidy up a paper quickly and efficiently. I would also like to thank Johannes Langguth as well as all the people from the Simula Laboratory for hosting me in Oslo for a week. It was a very rewarding experience for me to work in a different environment. I really hope to have the chance to collaborate with Bora, Fanny, Kamer, and Johannes in the future. I am proud to have been part of ROMA. The climate in the team is very welcoming and friendly. I very much enjoyed the lighthearted conversations we were having with the other PhD students in the team. They were always a nice break from the troubles of work. Hopefully we can all meet again in the future (and perhaps finally do a hike together). I am also thankful to Marie Bozo, Evelyne Blesle, Laetitia Lecot, Solene Audoux and Virginie Bouyer for helping me with the various administration tasks. I would like to thank Clémence Magnien and Alex Pothen for agreeing to review the manuscript of my thesis in detail and act as reporters in my defense. I am also grateful to Marthe Bonamy and Dimitrios Thilikos for agreeing to act as examinators in my defense. I have nothing but the biggest gratitude for my family and my sister, Katerina, in particular. I will never forget their support and how much they stood up for me. Knowing that I have so many people caring for me gave me all the strength I needed. To that end, I would like to dedicate this thesis in the memory of my parents and especially of my father. He supported me emotionally and financialy throughout all my studies and unfortunately he passed away before he could see me graduate. I would also like to thank all my friends both here in Lyon as well as those in Greece or other