Subgraph Counting in Subquadratic Time for Bounded Degeneracy Graphs

TL;DR

This paper introduces subquadratic algorithms for counting small subgraphs in bounded degeneracy graphs, advancing the understanding of efficient algorithms for this problem in real-world networks.

Contribution

It presents the first subquadratic algorithms for subgraph counting in bounded degeneracy graphs for patterns up to 9 vertices, and introduces a new reduction framework.

Findings

Subgraph counting for patterns up to 9 vertices can be done in rac{n^{5/3}} time.

Improved algorithms for counting cycles of length up to 10.

No prior subquadratic algorithms existed for these problems on bounded degeneracy graphs.

Abstract

We study the classic problem of subgraph counting, where we wish to determine the number of occurrences of a fixed pattern graph $H$ in an input graph $G$ of $n$ vertices. Our focus is on bounded degeneracy inputs, a rich family of graph classes that also characterizes real-world massive networks. Building on the seminal techniques introduced by Chiba-Nishizeki (SICOMP 1985), a recent line of work has built subgraph counting algorithms for bounded degeneracy graphs. Assuming fine-grained complexity conjectures, there is a complete characterization of patterns $H$ for which linear time subgraph counting is possible. For every $r \geq 6$, there exists an $H$ with $r$ vertices that cannot be counted in linear time. In this paper, we initiate a study of subquadratic algorithms for subgraph counting on bounded degeneracy graphs. We prove that when $H$ has at most $9$ vertices, subgraph counting can be done in $\tilde{O}(n^{5/3})$ time. As a secondary result, we give improved algorithms for counting cycles of length at most $10$. Previously, no subquadratic algorithms were known for the above problems on bounded degeneracy graphs. Our main conceptual contribution is a framework that reduces subgraph counting in bounded degeneracy graphs to counting smaller hypergraphs in arbitrary graphs. We believe that our results will help build a general theory of subgraph counting for bounded degeneracy graphs.