No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

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 − ɛ. J. Comput. System Sci. 74(3):335–349] proved that the problem is NP-hard to approximate within a factor 2 − ɛ, 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 − ɛ 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.