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