Subexponential algorithms for unique games and related problems

Subexponential time approximation algorithms are presented for the UNIQUE GAMES and SMALL-SET EXPANSION problems. Specifically, for some absolute constant c, the following two algorithms are presented. (1) An exp(knε)-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1- εc fraction of its constraints, outputs an assignment satisfying 1- ε fraction of the constraints. (2) An exp(nε/δ)-time algorithm that, given as input an n-vertex regular graph that has a set S of δn vertices with edge expansion at most εc, outputs a set S- of at most δn vertices with edge expansion at most ε. Subexponential algorithms are also presented with improved approximations to MAX CUT, SPARSEST CUT, and VERTEX COVER on some interesting subclasses of instances. These instances are graphs with low threshold rank, an interesting new graph parameter highlighted by this work. Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for UNIQUE GAMES. While the results here stop short of refuting the UGC, they do suggest that UNIQUE GAMES are significantly easier than NP-hard problems such as MAX 3-SAT, MAX 3-LIN, LABEL COVER, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every ε > 0 and every regular n-vertex graph G, by changing at most δ fraction of G's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most nε eigenvalues larger than 1- η, where η depends polynomially on ε. The subexponential algorithm combines this decomposition with previous algorithms for UNIQUE GAMES on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].