Parameterized (Modular) Counting and Cayley Graph Expanders

TL;DR

This paper provides a detailed complexity classification for counting specific subgraphs satisfying properties in large graphs, using novel Cayley graph expanders and modular counting techniques, advancing understanding in graph counting problems.

Contribution

It introduces new constructions of Cayley graph expanders of p-groups and applies them to achieve a comprehensive complexity classification for modular counting of subgraphs and homomorphisms.

Findings

Classified the complexity of counting edge-subgraphs with minor-closed properties.

Extended methods to modular counting of paths, cycles, forests, and matroid bases.

Provided a parameterized complexity classification for counting graph homomorphisms modulo a prime.

Abstract

We study the problem $\#\mathrm{EdgeSub}(\Phi)$ of counting $k$-edge subgraphs satisfying a given graph property $\Phi$ 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 $\mathbb{F}_p$ 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 $\#\mathrm{EdgeSub}(\Phi)$ for minor-closed properties $\Phi$, 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 investigate the problems of modular counting of paths, cycles, forests and matroid bases. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime $p$.