A threshold of ln n for approximating set cover

Abstract

We prove that (1 - o(l)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1-o(l))ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log2 n)/2 ≃ 0.72 ln n.