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.

翻译：我们考虑的是匹配的加权计算问题, 在任意的固定图形家庭 $\ textrm{ PrMatch} (mathcal{G}G}) 匹配时, 表示$\ textrrm{ g} 美元。 输入包括一个图形$\ mathcal{ G} 美元, 以及每个边缘的 $G$ 的合理的存在概率, 假设独立。 输出的概率是, 在结果分布中获得匹配$G$的概率。 也就是说, 一组对齐的边緣。 众所周知, 如果 $\ talfcal_ rcalcal_ talcal_ rcal_ talcaldr}, 则如果 $nclickral_ talcalal_ talth_ talth_ gratching} 美元, 则在复合时间里可以解析。 在本文中, 在某些假设下, 树维度的图形中, 也可以辨别所有图形家族 $_ 。