No Small Linear Program Approximates Vertex Cover within a Factor $2 - \epsilon$
Abstract
The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev (2003) proved that the problem is NP-hard to approximate within a factor , assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra (2002): vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomial size.
Cite
@article{arxiv.1503.00753,
title = {No Small Linear Program Approximates Vertex Cover within a Factor $2 - \epsilon$},
author = {Abbas Bazzi and Samuel Fiorini and Sebastian Pokutta and Ola Svensson},
journal= {arXiv preprint arXiv:1503.00753},
year = {2015}
}