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Parameterized (Modular) Counting and Cayley Graph Expanders

Authors Norbert Peyerimhoff, Marc Roth , Johannes Schmitt , Jakob Stix, Alina Vdovina

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Author Details

Norbert Peyerimhoff
  • Department of Mathematical Sciences, Durham University, UK
Marc Roth
  • Merton College, University of Oxford, UK
Johannes Schmitt
  • Mathematical Institute, University of Bonn, Germany
Jakob Stix
  • Mathematical Institute, Goethe-Universität Frankfurt, Germany
Alina Vdovina
  • School of Mathematics and Statistics, Newcastle University, UK

Funding

  • Schmitt, Johannes: The third author was supported by the SNF Early Postdoc.Mobility grant 184245 and thanks the Max Planck Institute for Mathematics in Bonn for its hospitality.

Acknowledgements

The second author is grateful to Holger Dell for fruitful discussions on the connections between [Radu Curticapean et al., 2021] and our work.

Cite AsGet BibTex

Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina. Parameterized (Modular) Counting and Cayley Graph Expanders. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 84:1-84:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.84

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Cayley graphs
  • counting complexity
  • expander graphs
  • fine-grained complexity
  • parameterized complexity

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