Subexponential Algorithms for d-to-1 Two-Prover Games and for Certifying Almost Perfect Expansion

A question raised by the recent subexponential algorithm for Unique Games (Arora, Barak, Steurer, FOCS 2010) is what other “hard-looking” problems admit good approximation algorithms with subexponential complexity. In this work, we give such an algorithm for d-to-1 two-prover games, a broad class of constraint satisfaction problems. Our algorithm has several consequences for Khot’s d-to-1 Conjectures. We also give a related subexponential algorithm for certifying that small sets in a graph have almost perfect expansion. Our algorithms follow the basic approach of the algorithms in (Arora, Barak, Steurer, FOCS 2010), but differ in the implementation of the individual steps. Key ingredients of our algorithms are a local version of Cheeger’s inequality that works in the regime of almost perfect expansion, and a graph decomposition algorithm that finds for every graph, a subgraph with at least an ε fraction of the edges such that every component has at most nO(log(1/λ)/ log(1/ε))1/2 eigenvalues larger than λ. ∗Microsoft Research New England, Cambridge, MA.