博士論文 Stable Matchings on Matroidal Structures (マトロイド的構造における安定マッチング) 横井 優 Stable Matchings on Matroidal Structures Yu Yokoi i Preface The stable matching model, introduced by Gale and Shapley (1962), is a mathematical for- mulation of two-sided markets, where each agent has a preference on the opposite agent set. Gale and Shapley provided an algorithm to find a stable matching, in which no agent has an incentive to change the current assignment. The algorithm has various applications such as the admission market between colleges and students and the labor market be- tween doctors and hospitals. As the growth of application, however, there arise various factors which cannot be handled by existing models. Then, the development of the stable matching theory shows no sign of slowing down even half a century after its appearance. In this thesis, we cope with some complicated settings of two-sided market models by extending the matroidal framework proposed by Fleiner (2001). We consider models with certain types of lower quotas, integer- or real- variables, and some complicated preferences. We capture these settings through matroids and extensions. Also, as a novel application of stable matchings, we solve a list coloring problem with supermodular constraints. We first investigate a stable matching model in which agents have lower quotas in addi- tion to upper quotas. This setting is well captured using generalized matroids. We provide a polynomial-time algorithm which finds a stable matching or reports the nonexistence. We also show properties of stable matchings such as the distributive lattice structure. Next, we consider stable allocations, real-variable versions of stable matchings. We design the first polynomial-time algorithm to find a stable allocation in polymatroid in- tersection. The algorithm combines the policy of the Gale{Shapley algorithm with the augmenting path technique, which is common in the matroid literature. Thirdly, we introduce a new notion \matroidal choice functions" to represent prefer- ences under matroid constraints, which cannot be derived from modular value functions. We find a strong relationship between the greedy algorithm on matroids and the substi- tutability of choice functions, an essential property for the existence of a stable matching. In the last part, utilizing a certain generalization of a stable matching, we prove the list supermodular coloring theorem, which generalizes Galvin's list edge coloring theorem for bipartite graphs. The existence of a stable matching plays a key role in the proof. ii Our research deals with different types of generalizations of the stable matching model. In all our models, matroidal structures work effectively when we design algorithms and analyze inherent properties. We can realize that many desirable results, such as the existence of polynomial-time algorithms and the lattice structure of stable matchings, are supported by simple axioms of matroidal structures. Since there are various kinds of two-sided markets in practice, it is important to estab- lish comprehensive guidelines to handle current and forthcoming problems. Our results suggest that matroidal structures can be useful tools for that purpose. iii Acknowledgments First and foremost, I would like to express my greatest thanks to my supervisor, Satoru Iwata. He supported me all the time in my Ph.D. course by providing right guidance and cheering words. Discussions with him were always full of novel ideas and helpful advice. His deep insight and comprehension on combinatorial optimization enhanced the significances of our results. Without his great support and patience, I could not lead such a happy life in my Ph.D. course. I am also grateful to my master's supervisor, Kazuo Murota. My interest to matroid theory and discrete convex analysis started when I was learning under him. He continu- ously provided me fruitful comments which came from his extensive knowledge. He also gave me pointed questions, which made my comprehension deeper. My thanks go to all current and former members of the laboratories, which I belonged to in my master and Ph.D. courses. It was so lucky for me that I was surrounded by kind and intelligent professors and colleagues. In particular, I thank Tasuku Soma, Yutaro Yamaguchi, Naoki Ito, and Tatsuya Matsuoka, with whom I spend a lot of time enjoying mathematical and daily conversations. Tasuku and Yutaro, my seniors, kindly taught me not only mathematical subjects but also how to submit a paper and how to apply for research grants. In particular, I am thankful to them for serving as good role models for me. To Naoki and Tatsuya, I am grateful for sharing their ideas, which were always thought- provoking for me. I also would like to acknowledge to Professors Kunihiko Sadakane, Akiko Takeda, Kazuhisa Makino, Hiroshi Hirai, Yuji Nakatsukasa, and Yusuke Kobayashi for their helpful comments in numerous seminars. I am deeply grateful to Professors Akihisa Tamura, Hiroshi Imai, Kunihiko Sadakane, and Hiroshi Hirai for reading this thesis carefully and providing lots of constructive advice. I also appreciate the financial support by JSPS Research Fellowship for Young Scien- tists, by JST ERATO Kawarabayashi Large Graph Project, and by JST CREST. Finally, I am grateful to my family and friends. With warm words, they always sup- ported and encouraged me to pursue my research as I wanted. v Contents 1 Introduction 1 1.1 The Basic Stable Matching Model ....................... 2 1.1.1 Model Formulation ............................ 2 1.1.2 The Deferred Acceptance Algorithm .................. 3 1.1.3 Structure of Stable Matchings ...................... 4 1.2 Stable Matchings and Matroidal Structures .................. 5 1.2.1 Matroidal Structures in Markets .................... 6 1.2.2 Preferences on Matroidal Structures .................. 7 1.2.3 Nonlinear Preferences on Matroidal Structures ............ 8 1.3 The Contribution ................................. 9 2 Preliminaries 13 2.1 Graphs ....................................... 13 2.2 Partial Orders ................................... 14 2.3 Matroids ...................................... 15 2.4 Submodular Functions and Polymatroids .................... 18 3 A Generalized Polymatroid Approach to Stable Matchings with Lower Quotas 21 3.1 Introduction .................................... 21 3.2 Generalized Matroids ............................... 23 3.2.1 Definition and Examples ......................... 23 3.2.2 Lower Extension of Generalized Matroids ............... 24 3.3 Preferences on Generalized Matroids ...................... 26 3.3.1 Dominance Relation ........................... 26 vi Contents 3.3.2 Choice Functions Induced by Ordered Matroids ............ 27 3.3.3 Matroid Kernels ............................. 28 3.4 Stable Matchings on Generalized Matroids ................... 29 3.4.1 Model Formulation ............................ 29 3.4.2 Characterization through Lower Extension .............. 29 3.4.3 Structure of Stable Matchings ...................... 30 3.4.4 Algorithm for Finding a Stable Matching ............... 31 3.5 Generalized Polymatroids ............................ 32 3.5.1 Definition ................................. 32 3.5.2 Lower Extension of Generalized Polymatroids ............. 34 3.6 Preferences on Generalized Polymatroids .................... 35 3.6.1 Optimal Points of Generalized Polymatroids .............. 35 3.6.2 Choice Functions Induced from Ordered Polymatroids ........ 36 3.7 Stable Allocations on Generalized Polymatroids ................ 39 3.7.1 Model Formulation ............................ 39 3.7.2 Choice Function Model ......................... 40 3.7.3 Characterization through Lower Extension .............. 40 3.7.4 Structure of Stable Allocations ..................... 41 Appendix 3.A Operations on Generalized Matroids ................ 42 Appendix 3.B Examples of Generalized Matroids ................. 44 Appendix 3.C Computations of Induced Choice Functions ............ 45 Appendix 3.D Supplements for Matroid Kernels .................. 49 4 Finding a Stable Allocation in Polymatroid Intersection 55 4.1 Introduction .................................... 55 4.2 Saturation and Exchange Capacities ...................... 60 4.2.1 Moving in Polymatroids ......................... 61 4.2.2 Preferences on Polymatroids ...................... 66 4.3 Algorithm ..................................... 67 4.4 Invariants ..................................... 70 4.5 Correctness .................................... 75 4.6 Complexity .................................... 77 4.7 Optimality .................................... 80 vii 5 Matroidal Choice Functions 87 5.1 Introduction .................................... 87 5.2 Matroidal Choice Functions ........................... 89 5.2.1 Definition ................................. 89 5.2.2 Representation with Circuit Families .................. 92 5.3 Algorithmic Characterization .......................... 94 5.4 Local Characterization .............................. 95 5.5 Relationships with Discrete Concavity ..................... 99 5.5.1 Valuated Matroids and M\-concave Functions ............. 99 5.5.2 Induced Matroidal Choice Functions .................. 100 5.5.3 Uninducible Matroidal Choice Function ................ 102 5.6 Matching Model with Matroidal Choice Functions .............. 104 5.6.1 Model Formulation ............................ 104 5.6.2 Matroidal Deferred Acceptance Algorithm ............... 105 5.6.3 Structure of Stable Matchings .....................