Three results towards the approximation of special maximum matchings in graphs

Abstract

For a graph $G$ define the parameters $\ell (G)$ and $L(G)$ as the minimum and maximum value of $\nu (G\backslash F)$, where $F$ is a maximum matching of $G$ and $\nu (G)$ is the matching number of $G$. In this paper, we show that there is a small constant $c>0$, such that the following decision problem is NP-complete: given a graph $G$ and $k\le \frac{|V|}{2}$, check whether there is a maximum matching $F$ in $G$, such that $|\nu (G\backslash F)-k|\le c\cdot |V|$. Note that when $c=1$, this problem is polynomial time solvable as we observe in the paper. Since in any graph $G$, we have $L(G)\le 2\ell (G)$, any polynomial time algorithm constructing a maximum matching of a graph is a 2-approximation algorithm for $\ell (G)$ and $\frac{1}{2}$-approximation algorithm for $L(G)$. We complement these observations by presenting two inapproximability results for $\ell (G)$ and $L(G)$.

14 pages, 9 figures. Revised according to the comments of referees. arXiv admin note: substantial text overlap with arXiv:2409.15388

References in corpus (4)

• On trees with a maximum proper partial 0-1 coloring containing a maximum matching
• On complexity of special maximum matchings constructing
• Two polynomial algorithms for special maximum matching constructing in trees
• An NP-hardness result for the colored constrained maximum 2-edge-colorable subgraph problem in bipartite graphs