Skip to Main content Skip to Navigation
Conference papers

Weighted Counting of Matchings in Unbounded-Treewidth 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 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.
Complete list of metadata

https://hal.telecom-paris.fr/hal-03712197
Contributor : Antoine Amarilli Connect in order to contact the contributor
Submitted on : Saturday, July 2, 2022 - 4:36:29 PM
Last modification on : Tuesday, August 2, 2022 - 3:08:50 AM

Links full text

Identifiers

  • HAL Id : hal-03712197, version 1
  • ARXIV : 2205.00851

Citation

Antoine Amarilli, Mikaël Monet. Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. MFCS, Aug 2022, Vienna, Austria. ⟨hal-03712197⟩

Share

Metrics

Record views

16