Weighted Counting of Matchings in Unbounded-Treewidth Graph Families

TL;DR

This paper establishes that weighted counting of matchings in graph families with unbounded treewidth is computationally intractable, extending known hardness results and characterizing bounded treewidth as the key to tractability.

Contribution

It proves that, under certain assumptions, bounded treewidth precisely characterizes the tractability of weighted matching counting in graph families.

Findings

Weighted counting of matchings is intractable for unbounded-treewidth graph families.

The result generalizes previous hardness results for special graph classes.

A similar intractability result applies to weighted counting of edge covers.

Abstract

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.