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).
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Khot, S., Popat, P., Saket, R. (2010). Approximate Lasserre Integrality Gap for Unique Games. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_23
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DOI: https://doi.org/10.1007/978-3-642-15369-3_23
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