Approximate Lasserre integrality gap for unique games

Abstract

In this paper, we investigate whether a constant round Lasserre Semi-definite Programming (SDP) relaxation might give a good approximation to the Unique Games problem. We show that the answer is negative if the relaxation is insensitive to a sufficiently small perturbation of the constraints. Specifically, we construct an instance of Unique Games with k labels along with an approximate vector solution to t rounds of the Lasserre SDP relaxation. The SDP objective is at least 1-ε whereas the integral optimum is at most γ, and all SDP constraints are satisfied up to an accuracy of δ>0. Here ε, γ>0 and t ∈ℤ+ are arbitrary constants and k=k(ε, γ) ∈ℤ+. The accuracy parameter δ can be made sufficiently small independent of parameters ε, γ, t, k (but the size of the instance grows as δ gets smaller). © 2010 Springer-Verlag.