A threshold of ln n for approximating set cover

Given a collection ℱ of subsets of S = 1, …, n, set cover is the problem of selecting as few as possible subsets from ℱ such that their union covers S, , and max k-cover is the problem of selecting k subsets from ℱ such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 - o (1) ) 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 alogorithm (which is (1 - o (1) ) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log 2 n) / 2 ≃0. 72 ln n. For max k -cover, we show an approximation threshold of (1 - 1/ e) (up to low-order terms), under assumption that P ≠ NP.