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The vertex cover problem is one of the most important and intensively studiedcombinatorial optimization problems. Khot and Regev (2003) proved that theproblem is NP-hard to approximate within a factor $2 - \epsilon$, assuming theUnique Games Conjecture (UGC). This is tight because the problem has an easy2-approximation algorithm. Without resorting to the UGC, the bestinapproximability 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 thefollowing unconditional result about linear programming (LP) relaxations of theproblem: every LP relaxation that approximates vertex cover within a factor$2-\epsilon$ has super-polynomially many inequalities. As a direct consequenceof our methods, we also establish that LP relaxations (as well as SDPrelaxations) that approximate the independent set problem within any constantfactor have super-polynomial size.

No Small Linear Program Approximates Vertex Cover within a Factor $2 - \epsilon$