Playing Random and Expanding Unique Games

In this work, we present a spectral algorithm that finds good assignments for instances of Unique Games when the underlying graph has some significant expansion and the constraints are arbitrary Γ-max-lin. We first analyze the behavior of the SDP by Feige and Lovász [FL92] on random instances of unique games. We show that on random d-regular graphs with permutations chosen at random, the value of the SDP is very small with probability 1− e−Ω(d). Hence, the SDP provides a proof of unsatisfiability for random unique games. We then give a spectral algorithm for recovering planted solutions. Given a random instance consistent with a given solution on 1 − fraction of the edges, our algorithm recovers a solution with value 1 − O( ) with high probability at least 1 − e−Ω(d) over the inputs. Using similar arguments as in the planted solution case, we conclude with an algorithm that finds good solutions for a Γ-max-lin expanding unique game. We present both cases in a unified manner in order to emphasize the main ideas that were used in the analysis of the algorithm.