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 = {e(1), e(2,)...,e(n)} en I and a collection F of subsets of E. The problem is to compute a sub-collection SOL subset of F such that boolean OR s(j epsilon SOL) S-j = E and its size ISOLI is minimized. When vertical bar S vertical bar <= k for all S E T we obtain. the unweighted k-set cover problem. It is well known that the greedy algorithm is an H-k-approximation algorithm for the unweighted k-set cover, where H-k = Sigma(k)(i=1) 1/j 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 (H-k - 1/2)-approximation algorithm. In this paper we present a new (H-k - 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.