Parameterized Results on Acyclic Matchings with Implications for Related Problems

TL;DR

This paper proves the computational hardness of approximating acyclic matchings in graphs, showing no fixed-parameter tractable approximation exists under common complexity assumptions, and also discusses kernelization limits for related parameters.

Contribution

It establishes FPT-inapproximability results for Acyclic Matching and related problems, and demonstrates kernelization lower bounds based on graph parameters.

Findings

No FPT-approximation within a constant factor assuming W[1] not in FPT.

FPT-inapproximability extends to Induced and Uniquely Restricted Matchings.

Acyclic Matching lacks polynomial kernels with respect to certain graph parameters unless NP is in coNP/poly.

Abstract

A matching $M$ in a graph $G$ is an \emph{acyclic matching} if the subgraph of $G$ induced by the endpoints of the edges of $M$ is a forest. Given a graph $G$ and a positive integer $\ell$, Acyclic Matching asks whether $G$ has an acyclic matching of size (i.e., the number of edges) at least $\ell$. In this paper, we first prove that assuming $\mathsf{W[1]\nsubseteq FPT}$, there does not exist any $\mathsf{FPT}$-approximation algorithm for Acyclic Matching that approximates it within a constant factor when the parameter is the size of the matching. Our reduction is general in the sense that it also asserts $\mathsf{FPT}$-inapproximability for Induced Matching and Uniquely Restricted Matching as well. We also consider three below-guarantee parameters for Acyclic Matching, viz. $\frac{n}{2}-\ell$, $\mathsf{MM(G)}-\ell$, and $\mathsf{IS(G)}-\ell$, where $n$ is the number of vertices in $G$, $\mathsf{MM(G)}$ is the matching number of $G$, and $\mathsf{IS(G)}$ is the independence number of $G$. Furthermore, we show that Acyclic Matching does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless $\mathsf{NP}\subseteq\mathsf{coNP}\slash\mathsf{poly}$.