Hardness of Approximation Between P and NP by Aviad Rubinstein A
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Hardness of Approximation Between P and NP by Aviad Rubinstein A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Christos Papadimitriou, Chair Professor Ilan Adler Associate Professor Prasad Raghavendra Professor Satish Rao Summer 2017 Hardness of Approximation Between P and NP Copyright 2017 by Aviad Rubinstein 1 Abstract Hardness of Approximation Between P and NP by Aviad Rubinstein Doctor of Philosophy in Computer Science University of California, Berkeley Professor Christos Papadimitriou, Chair Nash equilibrium is the central solution concept in Game Theory. Since Nash’s orig- inal paper in 1951, it has found countless applications in modeling strategic behavior of traders in markets, (human) drivers and (electronic) routers in congested networks, nations in nuclear disarmament negotiations, and more. A decade ago, the relevance of this solution concept was called into question by computer scientists [DGP09; CDT09], who proved (under appropriate complexity assumptions) that computing a Nash equilibrium is an intractable problem. And if centralized, specially designed algorithms cannot find Nash equilibria, why should we expect distributed, selfish agents to converge to one? The remaining hope was that at least approximate Nash equilibria can be efficiently computed. Understanding whether there is an efficient algorithm for approximate Nash equi- librium has been the central open problem in this field for the past decade. In this thesis, we provide strong evidence that even finding an approximate Nash equilibrium is intractable. We prove several intractability theorems for different settings (two- player games and many-player games) and models (computational complexity, query complexity, and communication complexity). In particular, our main result is that under a plausible and natural complexity assumption (“Exponential Time Hypoth- esis for PPAD”), there is no polynomial-time algorithm for finding an approximate Nash equilibrium in two-player games. The problem of approximate Nash equilibrium in a two-player game poses a unique technical challenge: it is a member of the class PPAD, which captures the complexity of several fundamental total problems, i.e. problems that always have a solution; and it also admits a quasipolynomial (≈ nlog n) time algorithm. Either property alone is believed to place this problem far below NP-hard problems in the complexity hierarchy; having both simultaneously places it just above P,atwhatcan 2 be called the frontier of intractability. Indeed, the tools we develop in this thesis to advance on this frontier are useful for proving hardness of approximation of several other important problems whose complexity lies between P and NP: Brouwer’s fixed point Given a continuous function f mapping a compact convex set to itself, Brouwer’s fixed point theorem guarantees that f has a fixed point, i.e. x such that f(x)=x. Our intractability result holds for the relaxed problem of finding an approximate fixed point, i.e. x such that f(x)≈x. Market equilibrium Market equilibrium is a vector of prices and allocations where the supply meets the demand for each good. Our intractability result holds for the relaxed problem of finding an approximate market equilibrium, where the supply of each good approximately meets the demand. CourseMatch (A-CEEI) Approximate Competitive Equilibrium from Equal In- come (A-CEEI) is the economic principle underlying CourseMatch, a system for fair allocation of classes to students (currently in use at Wharton, University of Pennsylvania). Densest k-subgraph Our intractability result holds for the following relaxation of the k-Clique problem: given a graph containing a k-clique, the algorithm has to find a subgraph over k vertices that is “almost a clique”, i.e. most of the edges are present. Community detection We consider a well-studied model of communities in social networks, where each member of the community is friends with a large fraction of the community, and each non-member is only friends with a small fraction of the community. VC dimension and Littlestone dimension The Vapnik-Chervonenkis (VC) di- mension is a fundamental measure in learning theory that captures the com- plexity of a binary concept class. Similarly, the Littlestone dimension is a measure of complexity of online learning. Signaling in zero-sum games We consider a fundamental problem in signaling, where an informed signaler reveals private information about the payoffs in a two-player zero-sum game, with the goal of helping one of the players. i ii Contents Contents ii List of Figures v List of Tables vi I Overview 1 1 The frontier of intractability 2 1.1 PPAD: Finding a needle you know isinthehaystack.......... 4 1.2 Quasi-polynomialtimeandthebirthdayparadox............ 11 1.3 Approximate Nash equilibrium ....................... 16 2 Preliminaries 18 2.1 Nash equilibrium and relaxations ...................... 18 2.2 PPAD and End-of-a-Line .......................... 20 2.3 Exponential Time Hypotheses ........................ 21 2.4 PCPtheorems................................. 21 2.5 Learning Theory ................................ 23 2.6 InformationTheory.............................. 24 2.7 Useful lemmata ................................. 26 II Communication Complexity 30 3 Communication Complexity of approximate Nash equilibrium 31 3.1 Proofoverview................................. 37 3.2 Proofs...................................... 41 3.3 An open problem: correlated equilibria in 2-player games ........ 57 iii 4 Brouwer’s fixed point 58 4.1 Brouwer with ∞ ............................... 58 4.2 Euclidean Brouwer ............................. 64 III PPAD 74 5 PPAD-hardness of approximation 75 6 The generalized circuit problem 78 6.1 Proofoverview................................. 80 6.2 From Brouwer to -Gcircuit ........................ 82 6.3 Gcrcuit withFan-out2........................... 98 7 Many-player games 101 7.1 Graphical, polymatrix games . ...................... 101 7.2 Succinctgames................................. 107 8 Bayesian Nash equilibrium 111 9 Market Equilibrium 114 9.1 Non-monotone markets: proof of inapproximability . ...... 118 10 Course Match 130 10.1TheCourseAllocationProblem....................... 132 10.2 A-CEEI is PPAD-hard............................. 134 10.3 A-CEEI ∈ PPAD ................................ 140 IV Quasi-polynomial Time 147 11 Birthday repetition 148 11.1 Warm-up: best -Nash.............................149 12 Densest k-Subgraph 154 12.1 Construction (and completeness) . .................... 158 12.2 Soundness . ................................... 159 13 Community detection 177 13.1 Hardness of counting communities . .................... 182 13.2 Hardness of detecting communities ..................... 184 iv 14 VC and Littlestone’s dimensions 188 14.1 Inapproximability of VC Dimension . .................... 192 14.2 Inapproximability of Littlestone’s Dimension . ............. 200 14.3Quasi-polynomialAlgorithmforLittlestone’sDimension........ 211 15 Signaling 214 15.1 Near-optimal signaling is hard ........................ 216 V Approximate Nash Equilibrium 222 16 2-Player approximate Nash Equilibrium 223 16.1 Technical overview ............................... 225 16.2 End-of-a-Line with local computation .................. 232 16.3 Holographic Proof ............................... 235 16.4PolymatrixWeakNash ............................ 248 16.5 From polymatrix to bimatrix ......................... 268 Bibliography 273 v List of Figures 4.1 AfacetoftheHirschetalconstruction..................... 63 4.2 Outsidethepicture................................. 64 4.3 GeometrynearaBrouwervertex........................ 71 6.1 Comparisonofaveraginggadgets........................ 81 14.1ReductionfromLabelCovertoVCDimension................ 212 14.2ReductionfromLabelCovertoLittlestone’sDimension........... 213 vi List of Tables 9.1 Goodsandtraders................................. 124 15.1VariablesinproofofTheorem15.1.1...................... 218 vii Acknowledgments I am incredibly lucky to have Christos as my advisor. I cannot compete with the praise written in dozens of Acknowledgment Sections in the dissertations of previous students of Christos. I can only attest that it’s all true. Christos, thank you for giving me beautiful problems to think about, for sprinkling your magic over my introductions, for the pre-deadline nights, and for showing me around Greece (in particular, Ikaria). Most importantly, thanks for all the advice! I also thank my thesis committee, Ilan Adler, Prasad Raghavendra, Satish Rao, and Christos. Their feedback during my qualifying exam was already tremendously helpful, and was my starting point for the last two parts of my thesis. For the past few years it has been a great pleasure to belong to the theory group at Berkeley. The lunches, retreats, basketball and soccer games, and occasional talks were an excellent inspiration. In particular, I was extremely fortunate to be part of the office: Jonah Brown-Cohen, Rishi Gupta, Alex Psomas, Tselil Schramm, Jarett Schwartz, Ning Tan, Ben Weitz, thanks! The latest perk of being a theorist at Berkeley is the Simons Institute. Each semester brings an influx of fascinating new and old visitors and learning oppor- tunities. Indeed, much of this thesis was first written on the white boards in the collaboration area. Thanks Dick Karp, Christos, Alistair Sinclair,