Three results towards the approximation of special maximum matchings in graphs

TL;DR

The paper investigates the extremal residual matching numbers $\ell(G)$ and $L(G)$, defined as the minimum and maximum of $\nu(G\setminus F)$ over all maximum matchings $F$ of a graph $G$. It proves that the additive-approximation problem for these parameters is NP-complete for any sublinear tolerance $f(|V|)$, while it is tractable when the tolerance scales linearly (e.g., $f(|V|)=|V|$ with a fixed constant). It further establishes multiplicative inapproximability results: under $P\neq NP$, there is no polynomial-time $(1-\varepsilon)$-approximation for $L(G)$ in connected bipartite graphs for $\varepsilon<1/88$ and no polynomial-time $(1+\varepsilon)$-approximation for $\ell(G)$ in connected bipartite graphs for $\varepsilon<1/80$, complemented by simple constant-factor guarantees from any maximum matching due to the bound $L(G)\le 2\ell(G)$. Together, these results map the hardness landscape for approximating residual matchings and motivate further study of approximation thresholds and tractable subclasses.

Abstract

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 $L(G)$.

Figures (9)

• Figure 1: A graph in which the maximum matching is not a subset of a maximum $2$-edge-colorable subgraph.
• Figure 2: $\ell(G)=1$ and $L(G)=2$ when $G$ is the path of length four.
• Figure 3: The gadget corresponding to the variable $x_i$ and the clause $C_j$.
• Figure 4: The gadget corresponding to the literal $\overline{x}_i$ and clause $C_j$.
• Figure 5: The conventional sign.

Theorems & Definitions (4)

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