Locally Searching for Large Induced Matchings

TL;DR

This paper analyzes the effectiveness of local search algorithms for finding large induced matchings across various graph classes, improving approximation ratios in certain cases.

Contribution

It extends the analysis of local search performance from regular graphs to broader graph classes, providing new approximation algorithms.

Findings

Improved approximation algorithms for induced matchings in specific graph classes.

Performance analysis of local search in $C_4$-free, $ ext{C}_3, ext{C}_4$-free, $C_5$-free, and claw-free graphs.

Generalization of existing results from regular to broader graph classes.

Abstract

It is an easy observation that a natural greedy approach yields a $\left(d-O(1)\right)$-factor approximation algorithm for the maximum induced matching problem in $d$-regular graphs. The only considerable and non-trivial improvement of this approximation ratio was obtained by Gotthilf and Lewenstein using a combination of the greedy approach and local search, where understanding the performance of the local search was the challenging part of the analysis. We study the performance of their local search when applied to general graphs, $C_4$-free graphs, $\{C_3,C_4\}$-free graphs, $C_5$-free graphs, and claw-free graphs. As immediate consequences, we obtain approximation algorithms for the maximum induced matching problem restricted to the $d$-regular graphs in these classes.