eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-08-22
9:1
9:15
10.4230/LIPIcs.MFCS.2022.9
article
Weighted Counting of Matchings in Unbounded-Treewidth Graph Families
Amarilli, Antoine
1
https://orcid.org/0000-0002-7977-4441
Monet, Mikaël
2
https://orcid.org/0000-0002-6158-4607
LTCI, Télécom Paris, Institut Polytechnique de Paris, France
Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol241-mfcs2022/LIPIcs.MFCS.2022.9/LIPIcs.MFCS.2022.9.pdf
Treewidth
counting complexity
matchings
Fibonacci sequence