No Small Linear Program Approximates Vertex Cover within a Factor 2 -- e

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [30], [31] proved that the problem is NP-hard to approximate within a factor 2 - ε, 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 in approximability result for the problem is due to Dinur and Safra [16], [17]: 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 of 2 - ε 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-polynomially many inequalities.