  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 $2 - \epsilon$, 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 (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
$2-\epsilon$ 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.
