A Characterization of Approximation Resistance
Abstract
A predicate f:{-1,1}^k -> {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.
Cite
@article{arxiv.1305.5500,
title = {A Characterization of Approximation Resistance},
author = {Subhash Khot and Madhur Tulsiani and Pratik Worah},
journal= {arXiv preprint arXiv:1305.5500},
year = {2013}
}
Comments
62 pages. The previous version of this paper gave a characterization of a modified notion called "Strong Approximation Resistance". We now present a characterization of approximation resistance