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oai:drops-oai.dagstuhl.de:14524 2024-03-06T10:54:14Z ddc:004 open_access

Parameterized (Modular) Counting and Cayley Graph Expanders Peyerimhoff, Norbert Roth, Marc Schmitt, Johannes Stix, Jakob Vdovina, Alina Cayley graphs counting complexity expander graphs fine-grained complexity parameterized complexity We study the problem #EdgeSub(Φ) of counting k-edge subgraphs satisfying a given graph property Φ in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field 𝔽_p which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(Φ) for minor-closed properties Φ, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Norbert Peyerimhoff and Marc Roth and Johannes Schmitt and Jakob Stix and Alina Vdovina 2021 Is Part Of LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2021.84 urn:nbn:de:0030-drops-145246 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.84 eng https://creativecommons.org/licenses/by/4.0/legalcode