Two Greedy Consequences for Maximum Induced Matchings

TL;DR

This paper presents new approximation algorithms for maximum induced matchings in specific classes of graphs, improving known performance ratios and providing bounds for k-degenerate graphs with bounded degree.

Contribution

It introduces an approximation algorithm with a performance ratio for $ ext{C}_3, ext{C}_5$-free $d$-regular graphs and establishes bounds for induced matchings in $k$-degenerate graphs.

Findings

Approximation ratio of 0.708...d+0.425 for certain graphs.

Every $k$-degenerate graph with maximum degree $d$ has an induced matching of size at least $m/((3k-1)d - k(k+1)+1)$.

Answers a previously open question about approximation ratios in specific graph classes.

Abstract

We prove that, for every integer $d$ with $d\geq 3$, there is an approximation algorithm for the maximum induced matching problem restricted to $\{ C_3,C_5\}$-free $d$-regular graphs with performance ratio $0.708\bar{3}d+0.425$, which answers a question posed by Dabrowski et al. (Theor. Comput. Sci. 478 (2013) 33-40). Furthermore, we show that every graph with $m$ edges that is $k$-degenerate and of maximum degree at most $d$ with $k<d$, has an induced matching with at least $m/((3k-1)d-k(k+1)+1)$ edges.