eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-31
34:1
34:17
10.4230/LIPIcs.ESA.2021.34
article
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
Curticapean, Radu
1
https://orcid.org/0000-0001-7201-9905
Dell, Holger
2
1
https://orcid.org/0000-0001-8955-0786
Husfeldt, Thore
1
3
https://orcid.org/0000-0001-9078-4512
Basic Algorithm Research Copenhagen (BARC), IT University of Copenhagen, Denmark
Goethe Universität Frankfurt, Germany
Lund University, Sweden
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol204-esa2021/LIPIcs.ESA.2021.34/LIPIcs.ESA.2021.34.pdf
Counting complexity
matchings
paths
subgraphs
parameterized complexity