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 $\mathsf{NP}\subseteq\mathsf{coNP}\slash\mathsf{poly}$.

翻译：在图$G$中，匹配$M$被称为**无环匹配**，若由$M$中边的端点诱导的$G$的子图是一个森林。给定图$G$和正整数$\ell$，无环匹配问题询问$G$是否包含大小（即边的数量）至少为$\ell$的无环匹配。本文首先证明：假设$\mathsf{W[1]\nsubseteq FPT}$，则当参数为匹配的大小时，不存在任何常数因子逼近的无环匹配$\mathsf{FPT}$-逼近算法。我们的归约具有一般性，它同样断言了诱导匹配和唯一受限匹配的$\mathsf{FPT}$-不可逼近性。我们还考虑了无环匹配的三个下界参数，即$\frac{n}{2}-\ell$、$\mathsf{MM(G)}-\ell$和$\mathsf{IS(G)}-\ell$，其中$n$是$G$的顶点数，$\mathsf{MM(G)}$是$G$的匹配数，$\mathsf{IS(G)}$是$G$的独立数。此外，我们证明除非$\mathsf{NP}\subseteq\mathsf{coNP}\slash\mathsf{poly}$，否则无环匹配在顶点覆盖数（或到团图的顶点删除距离）加上匹配大小的参数下不存在多项式核。