Parameterized results on acyclic matchings with implications for related problems

A matching M in a graph G is an acyclic matching if the subgraph of G induced by the endpoints of the edges of M is a forest. Given a graph G and $\ell \in N$, Acyclic Matching asks whether G has an acyclic matching of size at least ℓ. In this paper, we prove that assuming $W\left[1\right]⊈\mathrm{FPT}$, there does not exist any $\mathrm{FPT}$-approximation algorithm for Acyclic Matching that approximates it within a constant factor when parameterized by ℓ. Our reduction also asserts $\mathrm{FPT}$-inapproximability for Induced Matching and Uniquely Restricted Matching. We also consider three below-guarantee parameters for Acyclic Matching, viz. $\frac{n}{2}-\ell$, $\mathrm{MM}\left(G\right)-\ell$, and $\mathrm{IS}\left(G\right)-\ell$, where $n=V\left(G\right)$, $\mathrm{MM}\left(G\right)$ is the matching number, and $\mathrm{IS}\left(G\right)$ is the independence number of G. Also, 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 $\mathrm{NP}\subseteq \mathrm{coNP}/\mathrm{poly}$.