A Local Search-Based Approach for Set Covering

TL;DR

This paper introduces a novel non-oblivious local-search method for the Set Cover problem, achieving the first local-search-based approximation matching the theoretical bound, and improves approximation ratios for small set sizes.

Contribution

It presents the first local-search-based approximation for Set Cover matching the $H_k$ bound and refines previous bounds using larger moves and optimized potential functions.

Findings

First local-search-based $H_k$-approximation for Set Cover.

Improved approximation ratio by considering larger moves.

Provides an integrality gap result for the problem.

Abstract

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 $H_k \approx \ln k + O(1)$ 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 $(H_k - \Omega(1/k^8))$ (\emph{R.\ Hassin and A.\ Levin, SICOMP '05}) based on a modified greedy algorithm.