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 $\cal F$ of subsets of E. The problem is to compute a sub-collection SOL⊆$\cal F$ such that $\bigcup_{S_j\in SOL}S_j=E$ and its size |SOL| is minimized. When |S|≤k for all $S\in\cal F$ 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 $H_k=\sum_{i=1}^k {1 \over i}$ 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 $\left( H_k-{1\over 2}\right)$-approximation algorithm. In this paper we present a new $\left( H_k-{196\over 390}\right)$-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.