UCLA UCLA Electronic Theses and Dissertations Title Distributed Almost Stable Matchings Permalink https://escholarship.org/uc/item/6pr8d66m Author Rosenbaum, William Bailey Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Distributed Almost Stable Matchings A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by William Bailey Rosenbaum óþÕä © Copyright by William Bailey Rosenbaum óþÕä ABSTRACT OF THE DISSERTATION Distributed Almost Stable Matchings by William Bailey Rosenbaum Doctor of Philosophy in Mathematics University of California, Los Angeles, óþÕä Professor Rafail Ostrovsky, Chair e Stable Marriage Problem (SMP) is concerned with the follow scenario: suppose we have two disjoint sets of agents—for example prospective students and colleges, medical residents and hospitals, or potential romantic partners—who wish to be matched into pairs. Each agent has preferences in the form of a ranking of her potential matches. How should we match agents based on their preferences? We say that a matching is stable if no unmatched pair of agents mutually prefer each other to their assigned partners. In their seminal work on the SMP, Gale and Shapley [ÕÕ] prove that a stable matching exists for any preferences. ey further describe an ecient algorithm for nding a stable matching. In this dissertation, we consider the computational complexity of the SMP in the distributed setting, and the complexity of nding “almost sta- ble” matchings. Highlights include (Õ) communication lower bounds for nding stable and almost stable matchings, (ó) a distributed algorithm which nds an almost stable matching in polylog time, and (ì) hardness of approximation results for three dimensional analogues of the SMP. ii e dissertation of William Bailey Rosenbaum is approved. Marek Biskup Yiannis N Moschovakis Alexandr Sherstov Rafail Ostrovsky, Committee Chair University of California, Los Angeles óþÕä iii For Alivia—my stable partner iv Contents Acknowledgments vii Curriculum Vitae ix Õ Introduction Õ Õ.Õ eStableMarriageProblem ......................... ì Õ.ó Computational Complexity of the SMP . ÕÕ Õ.ì Distributed Stable Matchings . Õì Õ.¦ Almost Stable Matchings . Õß Õ.¢ ree Dimensional Stable Matching Problems . óÕ ó Communication Complexity ó¦ ó.Õ Communication Complexity Background . óÉ ó.ó Summary of Communication Results . ìþ ó.ì Lower Bounds for Exactly Stable Matchings . ìó ó.¦ General Proof of Main Results . ìß ì Approximation Algorithms ¦¢ ì.Õ Chapter Overview . ¦ß v ì.ó Preliminaries . ¦ ì.ì Deterministic Algorithm Description . ¢þ ì.¦ Performance Guarantees . ¢¢ ì.¢ Randomized Algorithms . äþ ì.ä Randomized Maximal and Almost Maximal Matchings . äì ¦ ree-gender Stable Matchings ä¢ ¦.Õ Chapter Overview . ä¢ ¦.ó Background and Denitions . äÉ ¦.ì Hardness of Approximation . ßì ¦.¦ Approximation Algorithms . þ ¦.¢ Concluding Remarks and Open Questions . ì ¦.ä Hardness of Gapc-3DM-3 ........................... ¦ A Computing Distance to Stability Éó A.Õ e Rotation Poset and Digraph . Éó A.ó Rotation Weights . ɦ A.ì Reduction to Max-Flow/Min-Cut . Éä Bibliography ÉÉ vi Acknowledgments First and foremost, I thank my advisor Ra Ostrovsky. ank you for pushing me to improve my results and nd the narrative in my arguments. I am awed and inspired by your keen insight and intuition. Your guidance has helped make me a better mathematician. I thank my doctoral committee members Marek Biskup, Yiannis Moschovakis, and Alexan- der Sherstov for their encouragement and support. I thank my collaborators, Yannai Gonczarowski and Noam Nisan who generously agreed to join forces when I learned we had been working independently on the same problem. e material in Chapter ó is based on our joint work [Õì]. I am especially grateful to Yannai for sharing his wisdom about the stable marriage problem, and LATEX dark magic. I have had the immense privilege of learning from some wonderful teachers throughout my life. I am especially thankful of (in roughly chronological order) Sherm Williamson, Linda Hill, Tom Wieting, Joe Roberts, Jerry Shurman, Benny Sudakov, and Alexander Sher- stov. ank you for your encouragement and your dedication to the art of teaching. I thank Olga Radko for helping me to become a better teacher. ank you to all of my students at UCLA—you are a perpetual reminder of why I started studying math in the rst place. ank you to my fellow mathematicians at UCLA, especially Sam Miner for being a wonderful hiking companion, pop culture connoisseur, and all-around great guy; Zach Nor- vii wood, Sudesh Kalyanswamy, and Andy Soer for being spectacular oce mates; Pietro Car- olino and Prabhanjan Ananth for sharing my excitement for TCS; and Jackie Lang for her tireless eorts to make the math department a better place. I thank the UCLA math department sta—especially Maggie Albert, Maida Bassili, Martha Contreras, and Leticia Dominguez—for helping me navigate my path through UCLA. I am eternally grateful to my parents, David and Margaret Rosenbaum, and my sister Liz Juusola. It has only been through your love and support that I have been able to ourish. ank you for always putting my education rst. I also thank the family I have gained since starting at UCLA: Patricia Novak, Nick Zappas, and Geri Novak. I cannot even express how helpful you have been these past few years. anks to Marika and Simona Zappas for being the middle and youngest sisters, respectively. ank you Alivia Zappas, for everything. I cannot imagine having done all of this with- out you. is one’s for you. Finally, I thank Finnegan Zappenbaum for reminding that sometimes the best thing is to go for a walk. viii Curriculum Vitae Education • University of California, Los Angeles, CA M.A. in Mathematics, Spring óþÕÕ • Reed College, Portland, OR B.A. in Mathematics, Spring óþþÉ Advisor: Jerry Shurman esis: Analysis on Circles: A Modern View of Fourier Series Published Work • Fast Distributed Almost Stable Matchings (with Rafail Ostrovsky), ACM Symposium on Principles of Distributed Computing (PODC), óþÕ¢. • A Stable Marriage Requires Communication (with Yannai Gonczarowski, Noam Nisan and Rafail Ostrovsky), ACM-SIAM Symposium on Discrete Algorithms (SODA), óþÕ¢. • It’s Not Easy Being ree: e Approximability of ree-Dimensional Stable Matching Problems (with Rafail Ostrovsky), International Workshop on Matching Under Pref- erences (MATCH-UP), óþÕ¢. ix • Analysis On Circles: A Modern View of Fourier Series. Undergraduate esis, Reed College, óþþÉ. Awards & Funding • Teaching Assistant Consultantship Department of Mathematics, UCLA, Fall óþÕ¢ • Student Travel Grant Association of Computing Machinery, PODC, óþÕ¢ • Graduate Student Instructorship Department of Mathematics, UCLA, óþÕ¦ • Robert Sorgenfrey Distinguished Teaching Award Department of Mathematics, UCLA, óþÕì • Phi Beta Kappa Reed College, óþþÉ • Commendation for Academic Achievement Reed College, óþþ¦–óþþ¢, óþþä–óþþß, óþþß–óþþ, óþþ–óþþÉ x Chapter Õ Introduction In this chapter we introduce the Stable Marriage Problem (SMP) which is the focus of this dissertation. e SMP was originally studied by Gale and Shapley in their seminal work, College Admissions and the Stability of Marriage in ÕÉäó [ÕÕ]. Since then, the SMP and its many variants—which we will refer to as stable matching problems—have been the focus of hundreds of journal articles and several books which span the elds of economics, dis- crete mathematics, and computer science. e broad appeal of stable matching problems is succinctly described by Kurt Melhorn in his forward to [ìÕ]: Matching under preferences is a topic of great practical importance, deep mathematical structure, and elegant algorithmics. e great practical impor- tance stems from the numerous applications such as the assignment of students to universities, families to housing, kidney transplant patients to donors, and so on. In these applications, the participants have preferences over the outcomes and the goal is to nd an assignment that optimizes the satisfaction of the par- ticipants. e importance of the area was recently clearly demonstrated to the Õ world-at-large by the award of the óþÕó Nobel Prize in Economics to Alvin Roth and Lloyd Shapley, partly for their work on the stable marriage problem. While stable matching problems are well studied, there remain many fascinating open questions of both theoretical and practical signicance. is dissertation provides new re- sults concerning the computational complexity of several variants of the SMP. We are es- pecially interested in distributed models of computation, where the input to a problem is scattered across several parties. While this model seems to be the natural context for many stable matching problems arising in nature, it has only been formally addressed in the liter- ature quite recently. e remainder of the chapter is organized as follows: Section Õ.Õ gives a formal deni- tion of the stable marriage problem, and describes the classical results of Gale and Shapley. It further describes an important variant of the SMP with incomplete preferences. Section Õ.ó gives results concerning the computational complexity of nding a stable matching. In Sec- tion Õ.ì we show that the SMP has a natural interpretation as a statement about graphs. We describe a computational model which allows us to consider the SMP as a distributed problem. We further prove a computational lower bound which