On the Complexity of Unique Games and Graph Expansion David Steurer A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance By the Department of Computer Science Advisor: Sanjeev Arora November 2010 © Copyright by David Steurer, 2010. All rights reserved. Abstract Understanding the complexity of approximating basic optimization prob- lems is one of the grand challenges of theoretical computer science. In recent years, a sequence of works established that Khot’s Unique Games Conjecture, if true, would settle the approximability of many of these problems, making this conjecture a central open question of the field. The results of this thesis shed new light on the plausibility of the Unique Games Conjecture, which asserts that a certain optimization problem, called Unique Games, is hard to approximate in a specific regime. On the one hand, we give the first confirmation of this assertion for a re- stricted model of computation that captures the best known approximation algorithms. The results of this thesis also demonstrate an intimate connection between the Unique Games Conjecture and approximability of graph expan- sion. In particular, we show that the Unique Games Conjecture is true if the expansion of small sets in graphs is hard to approximate in a certain regime. This result gives the first sufficient condition for the truth of the conjecture based on the inapproximability of a natural combinatorial problem. On the other hand, we develop efficient approximation algorithms for cer- tain classes of Unique Games instances, demonstrating that several previously proposed variants of the Unique Games Conjecture are false. Finally, we develop a subexponential-time algorithm for Unique Games, showing that this problem is significantly easier to approximate than NP-hard problems like Max 3-Sat, Max 3-Lin, and Label Cover, which are unlikely to have subexponential-time algorithm achieving a non-trivial approximation guaran- tee. This algorithm also shows that the inapproximability results based on the Unique Games Conjecture do not rule out subexponential-time algorithms, opening the possibility for such algorithms for many basic optimization prob- lems like Max Cut and Vertex Cover. At the heart of our subexponential-time algorithm for Unique Games lies a novel algorithm for approximating the expansion of graphs across different scales, which might have applications beyond Unique Games, especially in the design of divide-and-conquer algorithms. iii Acknowledgments I am greatly indebted to my advisor Sanjeev Arora, especially for his con- stant encouragement, his willingness to spent lots of time to discuss about research with me, and his efforts to improve my writing and presenting. His taste for problems, his commitment to research, and his scientific curiosity and creativity were a great source of inspiration. I couldn’t have wished for a better advisor. I am most thankful also to Boaz Barak, Prasad Raghavendra, and Nisheeth Vishnoi for their guidance, encouragement, and collaboration. They had considerable influence on the content of this thesis. I greatly enjoyed the many discussions with them about new problems and ideas. Many thanks to Sanjeev Arora, Boaz Barak, Moses Charikar, Bernard Chazelle, and Russell Impagliazzo for serving on my thesis committee. Over the years, I had the great pleasure to collaborate with many terrific researchers. I owe many thanks to my coauthors Sanjeev Arora, Boaz Barak, Markus Bläser, Constantinos Daskalakis, Benjamin Doerr, Venkatesan Gu- ruswami, Moritz Hardt, Ishay Haviv, Thomas Holenstein, Russell Impagliazzo, Subhash Khot, Alexandra Kolla, Johannes Lengler, Yury Makarychev, William Matthews, Prasad Raghavendra, Anup Rao, Oded Regev, Peter Sanders, Prasad Tetali, Madhur Tulsiani, Nisheeth Vishnoi, Avi Wigderson, and Yuan Zhou. I greatly enjoyed working with them and I have learned a lot from them. I thank all members of the theory groups in Princeton (at the University and at the Institute for Advanced Study) for creating such a great research environment. Thanks also to the members of the Center for Computational Intractability. The lively center meetings every month and the frequent workshops were great opportunities to learn and discuss about new results. Many thanks to the Algo group at the Labortoire de Recherche en Informa- tique (LRI) for hosting me during the winter of 2008. Many thanks also to Microsoft Research New England for hosting me as an intern in the summer of 2009. It was there that Prasad Raghavendra and I started working on graph expansion. Before coming to Princeton, I spent three terrific years at Saarland Univer- sity and the Max-Planck-Institute Informatik. Many thanks to all theorists there, especially my advisors Peter Sanders (now at Karlsruher Institute of Technology) and Kurt Mehlhorn. Special thanks to the competent and helpful administrative staff at the Computer Science Department of Princeton University, especially Mitra Kelly and Melissa Lawson. iv Heartfelt thanks to all my friends. They made the time in Princeton very enjoyable. Finally, I would like to thank my family — my parents Florence and Man- fred and my brother Hendrik — for their love and support. I gratefully acknowledge the National Science Foundation for their financial support. v Contents 1. Introduction1 1.1. Approximation Complexity....................2 1.2. Unique Games Conjecture.....................4 1.3. Graph Expansion..........................8 1.4. Small-Set Expansion Hypothesis.................. 10 1.5. Organization of this Thesis..................... 12 2. Background 16 2.1. Optimization and Approximation................. 16 2.2. Graphs, Expansion, and Eigenvalues............... 18 2.2.1. Proof of Local Cheeger Bound............... 21 2.3. Unique Games and Semidefinite Relaxation........... 23 2.4. Unique Games Conjecture and Reductions............ 27 I. Algorithms 34 3. Unique Games with Expanding Constraint Graphs 35 3.1. Main Result............................. 35 3.2. Propagation Rounding....................... 39 3.3. Tensoring Trick........................... 40 3.4. Notes................................. 45 4. Parallel Repeated Unique Games 47 4.1. Main Result............................. 47 4.2. Rounding Nonnegative Vectors.................. 49 4.3. From Arbitrary to Nonnegative Vectors.............. 53 4.3.1. Constructing Nearly Feasible Solutions.......... 55 4.3.2. Repairing Nearly Feasible Solutions........... 57 4.4. Notes................................. 62 vi Contents 5. Subexponential Approximation Algorithms 64 5.1. Main Results............................. 64 5.1.1. Small Set Expansion.................... 66 5.1.2. Unique Games........................ 67 5.2. Subspace Enumeration....................... 68 5.2.1. Small-Set Expansion.................... 70 5.2.2. Unique Games........................ 72 5.3. Threshold Rank vs. Small-Set Expansion............. 74 5.4. Low Threshold Rank Decomposition............... 76 5.5. Putting things together....................... 77 5.6. Notes................................. 78 II. Reductions 81 6. Graph Expansion and the Unique Games Conjecture 82 6.1. Main Results............................. 83 6.1.1. Proof of Theorem 6.2.................... 84 6.2. From Small-Set Expansion to Partial Unique Games ..... 84 6.2.1. Completeness........................ 85 6.2.2. Soundness.......................... 87 6.3. From Partial Unique Games to Unique Games ......... 94 6.3.1. Completeness........................ 96 6.3.2. Soundness.......................... 97 6.4. From Unique Games to Small-Set Expansion .......... 99 6.4.1. Completeness........................ 100 6.4.2. Soundness.......................... 101 6.5. Notes................................. 105 7. Reductions between Expansion Problems 106 7.1. Main Results............................. 106 7.2. Reduction............................... 108 7.2.1. Completeness........................ 110 7.2.2. Soundness.......................... 114 7.3. Notes................................. 117 vii Contents III. Lower Bounds 118 8. Limits of Semidefinite Relaxations 119 8.1. Overview............................... 119 8.1.1. Techniques.......................... 122 8.1.2. Organization......................... 123 8.2. Proof Sketch............................. 123 8.3. Hierarchies of SDP Relaxations.................. 128 8.3.1. LHr-Relaxation........................ 128 8.3.2. SAr-Relaxation........................ 129 8.3.3. Robustness.......................... 130 8.4. Weak Gaps for Unique Games ................... 130 8.4.1. Properties of Weak Gap Instances............. 131 8.5. Integrality Gap Instance for Unique Games ........... 134 8.6. Integral Vectors........................... 134 8.7. Local Distributions for Unique Games .............. 140 8.8. Construction of SDP Solutions for E2LIN(q)........... 142 8.8.1. Vector Solution....................... 143 8.8.2. Local Distributions..................... 144 8.9. Smoothing.............................. 150 8.10. Notes................................. 156 9. Open Problems 158 Bibliography 160 viii 1. Introduction A central goal of theoretical computer science is to understand the complexity of basic computational tasks, that is, the minimum amount of computational resources (especially, time) needed to carry out such tasks. We will focus on computational tasks arising from combinatorial optimiza- tion problems. Here, the goal is to optimize a given objective function