Hardness of Approximation Between P and NP

313 Pages·2017·1.66 MB·English

Hardness of Approximation Between P and NP
Aviad Rubinstein
Electrical Engineering and Computer Sciences
University of California at Berkeley
Technical Report No. UCB/EECS-2017-146
http://www2.eecs.berkeley.edu/Pubs/TechRpts/2017/EECS-2017-146.html
August 11, 2017
Copyright © 2017, by the author(s).
All rights reserved.

Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission.
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
The dissertation of Aviad Rubinstein, titled Hardness of Approximation Between
P and NP, is approved:
Chair Date
Date
Date
Date
University of California, Berkeley
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
oftradersinmarkets, (human)driversand(electronic)routersincongestednetworks,
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, weprovidestrongevidencethatevenfindinganapproximateNashequilibrium
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 ( nlogn) 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, at what can
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. xsuchthatf x x. Ourintractabilityresultholdsfortherelaxedproblem
( )=
of finding an approximate fixed point, i.e. x such that f x x.
( )≈
Market equilibrium Marketequilibriumisavectorofpricesandallocationswhere
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
forfairallocationofclassestostudents(currentlyinuseatWharton, 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 is in the haystack . . . . . . . . . . 4
1.2 Quasi-polynomial time and the birthday paradox . . . . . . . . . . . . 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 PCP theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Learning Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Useful lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
II Communication Complexity 30
3 Communication Complexity of approximate Nash equilibrium 31
3.1 Proof overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (cid:96) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
∞
4.2 Euclidean Brouwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
III PPAD 74
5 PPAD-hardness of approximation 75
6 The generalized circuit problem 78
6.1 Proof overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 From Brouwer to (cid:15)-Gcircuit . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 Gcrcuit with Fan-out 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Many-player games 101
7.1 Graphical, polymatrix games . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Succinct games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8 Bayesian Nash equilibrium 111
9 Market Equilibrium 114
9.1 Non-monotone markets: proof of inapproximability . . . . . . . . . . . 118
10 Course Match 130
10.1 The Course Allocation Problem . . . . . . . . . . . . . . . . . . . . . . . 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 (cid:15)-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