On the Complexity of Local Search for Weighted Standard Set Problems

TL;DR

This paper investigates the computational complexity of finding locally optimal solutions in weighted standard set problems, revealing PLS-completeness for many cases and providing tight bounds for specific problems.

Contribution

It establishes PLS-completeness for local search in weighted SetCover and SetPacking with simple neighborhoods, a rare result in this domain.

Findings

Local search for weighted SetCover and SetPacking is PLS-complete with neighborhood size one.

Tight bounds are derived for neighborhoods of size two in these problems.

This work provides some of the first PLS complexity results for weighted standard set problems.

Abstract

In this paper, we study the complexity of computing locally optimal solutions for weighted versions of standard set problems such as SetCover, SetPacking, and many more. For our investigation, we use the framework of PLS, as defined in Johnson et al., [JPY88]. We show that for most of these problems, computing a locally optimal solution is already PLS-complete for a simple neighborhood of size one. For the local search versions of weighted SetPacking and SetCover, we derive tight bounds for a simple neighborhood of size two. To the best of our knowledge, these are one of the very few PLS results about local search for weighted standard set problems.