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

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 γ > 0, such that there is no o(m 1+γ ) time algorithm for counting H-homomorphisms.

Counting Homomorphic Cycles in Degenerate Graphs

“…Specifically, they showed that H-homomorphisms in bounded-degeneracy graphs can be counted in time Õ(n) if and only if H contains no induced cycles of length 6 or larger 6 . A similar result has been independently obtained in [7]. These results highlight the important role played by cycles of length at least 6, as being the minimal graphs whose homomorphisms cannot be counted in (almost) linear time (in bounded-degeneracy graphs).…”

“…To establish that this is indeed the "hardest orientation", we show (roughly speaking) that for every DAG H and for every directed subdivision H ′ of H, counting H ′homomorphisms can be reduced to counting H-homomorphisms (in input DAGs of bounded maximum degree). It is easy to see that every acyclic orientation H of C ℓ (with at least two sources) is a directed subdivision of the alternating orientation of C ℓ ′ for some even ℓ ′ ≤ ℓ, and that ℓ ′ = ℓ if and only if H itself is alternating 7 . It follows that the running time of counting C ℓ -homomorphisms is dominated by the running time of counting homomorphisms of alternating orientations of cycles of even length at most ℓ.…”

“…The study of subgraph counting in degenerate graphs goes back to a classical work of Chiba and Nishizeki [10] from the early 1980s. Very recently, this research direction has seen several substantial developments [6,7,9,18]. Bressan [9] gave a general algorithm for counting homomorphisms in bounded-degeneracy graphs, which in particular implies a sufficient condition 5 (on graphs H) under which H-homomorphisms can be counted in time Õ(n).…”