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TR13-146 | 20th October 2013 07:35

#### A Characterization of Approximation Resistance

TR13-146
Authors: Subhash Khot, Madhur Tulsiani, Pratik Worah
Publication: 22nd October 2013 10:16
Keywords:

Abstract:

A predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$ is called {\it approximation resistant} if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment that satisfies at least $\rho(f)+\Omega(1)$ fraction of the constraints.

We present a complete characterization of approximation resistant predicates under the Unique Games Conjecture. We also present characterizations in the {\it mixed} linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy. In the former case, the characterization coincides with the one based on UGC. Each of the two characterizations is in terms of existence of a probability measure with certain symmetry properties on a natural convex polytope associated with the predicate.

This is a revised version of out previous paper which gave a characterization for a modified notion called "Strong Approximation Resistance".

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