Approximate Lasserre Integrality Gap For Unique Games

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 - epsilon whereas the integral optimum is at most gamma, and all SDP constraints are satisfied up to an accuracy of delta > 0. Here epsilon, gamma > 0 and t is an element of Z(+) are arbitrary constants and k = k(epsilon, gamma) is an element of Z(+). The accuracy pararneter delta can be made sufficiently small independent of parameters epsilon, gamma, t, k (but the size of the instance grows as 6 gets smaller).