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Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths

Authors Radu Curticapean , Holger Dell , Thore Husfeldt

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Author Details

Radu Curticapean
  • Basic Algorithm Research Copenhagen (BARC), IT University of Copenhagen, Denmark
Holger Dell
  • Goethe Universität Frankfurt, Germany
  • Basic Algorithm Research Copenhagen (BARC), IT University of Copenhagen, Denmark
Thore Husfeldt
  • Basic Algorithm Research Copenhagen (BARC), IT University of Copenhagen, Denmark
  • Lund University, Sweden

Funding

Supported by VILLUM Foundation grant 16582.

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Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.34

Abstract

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of k-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an n^{f(t,s)}-time algorithm to compute modulo 2^t the number of subgraph occurrences of patterns that are s vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo 2^t. Complementing our algorithm, we also give a simple and self-contained proof that counting k-matchings modulo odd integers q is {Mod}_q W[1]-complete and prove that counting k-paths modulo 2 is ⊕W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Counting complexity
  • matchings
  • paths
  • subgraphs
  • parameterized complexity

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