Weighted Counting of Matchings in UnboundedTreewidth Graph Families
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 treewidthconstructibility 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 treewidthconstructible 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, 3regular, 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.00851
 arXiv:
 arXiv:2205.00851
 Bibcode:
 2022arXiv220500851A
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics
 EPrint:
 This is the full version with proofs of the MFCS'22 article