A Local Search-Based Approach for Set Covering

In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a H-k approximate to ln k+ O(1) approximation for this problem, where the size of each set is bounded by k. Moreover, getting a ln k-O(ln ln k) approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of k, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first H-k - approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an (H-k - Omega(log(2) k)/k)-approximation, improving on the previous bound of (H-k - Omega(1/k(8))) (R. Hassin and A. Levin, SICOMP '05) based on a modified greedy algorithm.