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

In the unweighted set cover problem we are given a set of elements $E=\{e_1,e_2,\ldots,e_n\}$ and a collection ${\cal F}$ of subsets of E. The problem is to compute a subcollection $SOL\subseteq{\cal F}$ such that $\bigcup_{S_j\in SOL}S_j=E$ and its size $|SOL|$ is minimized. When $|S|\leq k$ for all $S\in{\cal F}$, we obtain the unweighted k-set cover problem. It is well known that the greedy algorithm is an $H_k$-approximation algorithm for the unweighted k-set cover, where $H_k=\sum_{i=1}^k\frac{1}{i}$ is the kth 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 $(H_k-\frac{1}{2})$-approximation algorithm. In this paper we present a new $(H_k-\frac{196}{390})$-approximation algorithm for $k\geq4$ that improves the previous best approximation ratio for all values of $k\geq4$. Our algorithm is based on combining a local search during various stages of the greedy algorithm.