Three results towards the approximation of special maximum matchings in graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912348166422528 |
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| author | Mkrtchyan, Vahan |
| author_facet | Mkrtchyan, Vahan |
| contents | For a graph $G$ define the parameters $\ell(G)$ and $L(G)$ as the minimum and maximum value of $ν(G\backslash F)$, where $F$ is a maximum matching of $G$ and $ν(G)$ 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 $k\leq \frac{|V|}{2}$, check whether there is a maximum matching $F$ in $G$, such that $|ν(G\backslash F)-k|\leq c\cdot |V|$. Note that when $c=1$, this problem is polynomial time solvable as we observe in the paper. Since in any graph $G$, we have $L(G)\leq 2\ell(G)$, 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 $L(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16324 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three results towards the approximation of special maximum matchings in graphs Mkrtchyan, Vahan Combinatorics 05C85, 68R10, 05C70, 05C15 For a graph $G$ define the parameters $\ell(G)$ and $L(G)$ as the minimum and maximum value of $ν(G\backslash F)$, where $F$ is a maximum matching of $G$ and $ν(G)$ 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 $k\leq \frac{|V|}{2}$, check whether there is a maximum matching $F$ in $G$, such that $|ν(G\backslash F)-k|\leq c\cdot |V|$. Note that when $c=1$, this problem is polynomial time solvable as we observe in the paper. Since in any graph $G$, we have $L(G)\leq 2\ell(G)$, 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 $L(G)$. |
| title | Three results towards the approximation of special maximum matchings in graphs |
| topic | Combinatorics 05C85, 68R10, 05C70, 05C15 |
| url | https://arxiv.org/abs/2409.16324 |