Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

TL;DR

This paper characterizes exactly when near-linear time algorithms exist for counting homomorphisms of small pattern graphs in bounded degeneracy graphs, revealing a sharp complexity dichotomy based on the pattern's induced cycle length.

Contribution

It provides a complete dichotomy for the complexity of homomorphism counting in bounded degeneracy graphs based on the pattern's induced cycle length, resolving an open problem.

Findings

Algorithms exist for patterns with induced cycles of length ≤ 5.

No near-linear algorithms are possible for patterns with induced cycles ≥ 6, assuming standard conjectures.

The complexity boundary is precisely characterized by the length of the largest induced cycle in the pattern graph.

Abstract

Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and the pattern $H$. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of $H$, $H$-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs. Can we precisely characterize the patterns $H$ for which near-linear time algorithms are possible? We completely resolve this problem, discovering a clean dichotomy using fine-grained complexity. Let $m$ denote the number of edges in $G$. We prove the following: if the largest induced cycle in $H$ has length at most $5$, then there is an $O(m\log m)$ algorithm for counting $H$-homomorphisms in bounded degeneracy graphs. If the largest induced cycle in $H$ has length at least $6$, then (assuming standard fine-grained complexity conjectures) there is a constant $\gamma > 0$, such that there is no $o(m^{1+\gamma})$ time algorithm for counting $H$-homomorphisms.