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

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.