Approximating the Unweighted k-Set Cover Problem: Greedy Meets Local Search

In the unweighted set-cover problem we are given a set of elements E = {e1, e2, . . . , en} and a collection F of subsets of E. The problem is to compute a sub-collection SOL ⊆F such that ⋃ Sj∈SOL Sj = E and its size |SOL| is minimized. When |S| ≤ k for all S ∈ F we obtain the unweighted k-set cover problem. It is well known that the greedy algorithm is an Hk-approximation algorithm for the unweighted k-set cover, where Hk = ∑k i=1 1 i is the k-th harmonic number, and that this bound on the approximation ratio of the greedy algorithm, is tight for all constant values of k. Since the set cover problem is a fundamental problem, there is an ongoing research effort to improve this approximation ratio using modifications of the greedy algorithm. The previous best improvement of the greedy algorithm is an ( Hk − 12 ) -approximation algorithm. In this paper we present a new ( Hk − 196 390 ) -approximation algorithm for k ≥ 4 that improves the previous best approximation ratio for all values of k ≥ 4. Our algorithm is based on combining local search during various stages of the greedy algorithm.