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A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs

Authors Daniel Paul-Pena , C. Seshadhri

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Homomorphism counting
  • Bounded degeneracy graphs
  • Fine-grained complexity
  • Orbit counting
  • Subgraph counting

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Abstract

Counting the number of homomorphisms of a pattern graph H in a large input graph G is a fundamental problem in computer science. In many applications in databases, bioinformatics, and network science, we need more than just the total count. We wish to compute, for each vertex v of G, the number of H-homomorphisms that v participates in. This problem is referred to as homomorphism orbit counting, as it relates to the orbits of vertices of H under its automorphisms.
Given the need for fast algorithms for this problem, we study when near-linear time algorithms are possible. A natural restriction is to assume that the input graph G has bounded degeneracy, a commonly observed property in modern massive networks. Can we characterize the patterns H for which homomorphism orbit counting can be done in near-linear time?
We discover a dichotomy theorem that resolves this problem. For pattern H, let 𝓁 be the length of the longest induced path between any two vertices of the same orbit (under the automorphisms of H). If 𝓁 ≤ 5, then H-homomorphism orbit counting can be done in near-linear time for bounded degeneracy graphs. If 𝓁 > 5, then (assuming fine-grained complexity conjectures) there is no near-linear time algorithm for this problem. We build on existing work on dichotomy theorems for counting the total H-homomorphism count. Surprisingly, there exist (and we characterize) patterns H for which the total homomorphism count can be computed in near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

Cite As Get BibTex

Daniel Paul-Pena and C. Seshadhri. A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.54

Author Details

Daniel Paul-Pena
  • University of California, Santa Cruz, CA, USA
C. Seshadhri
  • University of California, Santa Cruz, CA, USA

Funding

Both authors are supported by NSF CCF-1740850, CCF-1839317, CCF-2402572, and DMS-2023495.

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