A Dichotomy Hierarchy for Linear Time Subgraph Counting in Bounded Degeneracy Graphs
Year: 2026
Type: article
Abstract: Subgraph and homomorphism counting are fundamental algorithmic problems. Given a constant-size pattern graph H and a large input graph G , we wish to count the number of H -homomorphisms/subgraphs in G . Given the massive sizes of real-world graphs and the practical importance of counting problems, we focus on studying when (near) linear-time algorithms are possible. The seminal work of Chiba-Nishizeki (SICOMP 1985) shows that for bounded-degeneracy input classes, clique and 4-cycle counting can be done in linear time. Recent works (Bera et al, SODA 2021, JACM 2022) show a dichotomy theorem characterizing the patterns H for which H -homomorphism counting is possible in linear time, for bounded-degeneracy inputs. At the other end, NeΕ‘etΕil and Ossona de Mendez used their deep theory of βsparsityβ to define graph classes with bounded expansion (which contains all non-trivial minor-closed families). They prove that, for all H , H -homomorphism counting can be done in linear time for bounded expansion inputs. What lies between? For a specific H , can we characterize input classes where H -homomorphism counting is possible in linear time? We discover a hierarchy of dichotomy theorems that answer the above questions. We show the existence of an infinite sequence of families of graph classes π’ 0 β π’ 1 β β¦ β π’ β where π’ 0 is the family of bounded-degeneracy classes, and π’ β is the family of bounded expansion classes. Fix any constant-size pattern graph H . Let LICL(H) denote the length of the longest induced cycle in H . We prove the following. If LICL(H) < 3(r+2), then H -homomorphisms can be counted in linear time for inputs in π’ r . If LICL(H) β₯ 3( r +2), then (assuming fine-grained complexity conjectures) H -homomorphism counting on inputs from π’ r takes π ( n 1+π ) time. (Here, n denotes the number of vertices of the input, and π is some explicit constant.) Similar dichotomy theorems hold for subgraph counting.
Source: Journal of the ACM
DOI: 10.1145/3848505
Authors: Daniel Paul-Pena, C. Seshadhri
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Funder: National Science Foundation
OpenAlex ID: W7213450324