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

TL;DR

This paper proves that any linear programming relaxation approximating vertex cover within a factor close to 2 must have super-polynomial size, highlighting fundamental limits of LP approaches for this problem.

Contribution

It establishes that LP relaxations achieving near-2 approximation for vertex cover require super-polynomial size, providing an unconditional complexity barrier.

Findings

Any LP approximation within 2 - ε has super-polynomial size.

LP and SDP relaxations approximating independent set within any constant factor are super-polynomial.

The result is unconditional, not relying on the Unique Games Conjecture.

Abstract

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.