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

TL;DR

This paper explores the computational complexity of counting subgraph patterns modulo fixed integers, providing algorithms for certain cases and proving hardness results for others, advancing understanding of modular subgraph counting.

Contribution

It generalizes polynomial-time algorithms for counting matchings modulo powers of two and establishes W[1]-hardness for counting matchings and paths modulo odd integers and two, respectively.

Findings

Polynomial-time algorithm for counting subgraph patterns modulo 2^t

W[1]-hardness of counting k-matchings modulo odd integers q

W[1]-hardness of counting k-paths modulo 2

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).