Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths

Radu Curticapean; Holger Dell; Thore Husfeldt

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arXiv · 2107.00629

Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths

Radu Curticapean·Holger Dell·Thore Husfeldt

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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 Parity-W[1]-complete, answering an open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).

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BibTeX RIS

Radu Curticapean, Holger Dell, Thore Husfeldt. 2021-07-01. Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths. https://arxiv.org/abs/2107.00629

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