  We consider a weighted counting problem on matchings, denoted
$\textrm{PrMatching}(\mathcal{G})$, on an arbitrary fixed graph family
$\mathcal{G}$. The input consists of a graph $G\in \mathcal{G}$ and of rational
probabilities of existence on every edge of $G$, assuming independence. The
output is the probability of obtaining a matching of $G$ in the resulting
distribution, i.e., a set of edges that are pairwise disjoint. It is known
that, if $\mathcal{G}$ has bounded treewidth, then
$\textrm{PrMatching}(\mathcal{G})$ can be solved in polynomial time. In this
paper we show that, under some assumptions, bounded treewidth in fact
characterizes the tractable graph families for this problem. More precisely, we
show intractability for all graph families $\mathcal{G}$ satisfying the
following treewidth-constructibility requirement: given an integer $k$ in
unary, we can construct in polynomial time a graph $G \in \mathcal{G}$ with
treewidth at least $k$. Our hardness result is then the following: for any
treewidth-constructible graph family $\mathcal{G}$, the problem
$\textrm{PrMatching}(\mathcal{G})$ is intractable. This generalizes known
hardness results for weighted matching counting under some restrictions that do
not bound treewidth, e.g., being planar, 3-regular, or bipartite; it also
answers a question left open in Amarilli, Bourhis and Senellart (PODS'16). We
also obtain a similar lower bound for the weighted counting of edge covers.

Computational Complexity;Discrete Mathematics