Counting Subgraphs in Somewhere Dense Graphs

TL;DR

This paper classifies the fixed-parameter tractability of counting small subgraphs like matchings and independent sets in large graphs, based on the structural properties of the host graph classes, revealing clear complexity dichotomies.

Contribution

It provides exhaustive complexity classifications for counting problems in restricted graph classes, especially distinguishing between nowhere dense and somewhere dense classes.

Findings

Counting k-matchings is FPT iff the class is nowhere dense.

Counting k-independent sets is FPT iff the class is nowhere dense.

Almost tight lower bounds are established for somewhere dense classes.

Abstract

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time $f(H)\cdot |G|^{O(1)}$ for some computable function $f$. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes $\mathcal{G}$ as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting $k$-matchings in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. (2) Counting $k$-independent sets in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if $\mathcal{G}$ is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting $k$-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in $F$-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting $k$-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).