Approximate Lasserre integrality gap for unique games

Authors:
Subhash Khot;Preyas Popat;Rishi Saket
Affiliations:
Computer Science Department, New York University;Computer Science Department, New York University;Computer Science Department, Carnegie Mellon University
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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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 bemade sufficiently small independent of parameters ε, γ,t, k (but the size of the instance grows as δ gets smaller).