No Small Linear Program Approximates Vertex Cover Within a Factor 2-epsilon

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [Khot S, Regev O (2008) Vertex cover might be hard to approximate to within 2 - epsilon. J. Comput. System Sci. 74(3): 335-349] proved that the problem is NP-hard to approximate within a factor 2- epsilon, 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 [Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann. Math. 162(1):439-485]: 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 the vertex cover within a factor 2 - epsilon has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as semidefinite programming relaxations) that approximate the independent set problem within any constant factor have a super-polynomial size.