Weighted Counting of Matchings in Unbounded-Treewidth Graph Families

Authors Antoine Amarilli , Mikaël Monet

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Antoine Amarilli
  • LTCI, Télécom Paris, Institut Polytechnique de Paris, France
Mikaël Monet
  • Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France


We thank Florent Capelli and Sébastien Tavenas for discussions on the problem, in particular on how to prove Proposition 5.2.

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Antoine Amarilli and Mikaël Monet. Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We consider a weighted counting problem on matchings, denoted PrMatching(𝒢), on an arbitrary fixed graph family 𝒢. The input consists of a graph G ∈ 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 𝒢 has bounded treewidth, then PrMatching(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 𝒢 satisfying the following treewidth-constructibility requirement: given an integer k in unary, we can construct in polynomial time a graph G ∈ 𝒢 with treewidth at least k. Our hardness result is then the following: for any treewidth-constructible graph family 𝒢, the problem PrMatching(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 et al., 2016]. We also obtain a similar lower bound for the weighted counting of edge covers.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Treewidth
  • counting complexity
  • matchings
  • Fibonacci sequence


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