Locally Searching for Large Induced Matchings

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.