Approximating Unique Games Using Low Diameter Graph Decomposition

We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the Max2Lin_k problem in K_r-minor free graphs, when there is an assignment satisfying 1-ε fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(rε) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the MaxCut problem for K_r-minor free graphs. For general Unique Games in K_r-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r √(ε)) fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the Max2Lin_k problem and by √(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the Max2Lin_k case, and the latter result generalizes the result of Trevisan for general Unique Games.