Approximation Algorithms for Unique Games ∗ Luca Trevisan
نویسندگان
چکیده
A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC’02) conjectured that for arbitrarily small γ,ε > 0 it is NPhard to distinguish games of value smaller than γ from games of value larger than 1− ε . Several recent inapproximability results rely on Khot’s conjecture. Considering the case of sub-constant ε , Khot (STOC’02) analyzes an algorithm based on semidefinite programming that satisfies a constant fraction of the constraints in unique games of value 1−O(k−10 · (logk)−5), where k is the size of the domain of the variables. We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1−O(1/ logn), satisfies a constant fraction of the constraints, where n is the number of variables. This is an improvement over Khot’s algorithm if the domain is sufficiently large. ∗An extended abstract of this paper appeared in the Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pages 197–205. †This material is based upon work supported by the National Science Foundation under grant No. CCF 0515231 and by the BSF under grant 2002246. ACM Classification: F.2.2 AMS Classification: 68Q17
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