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Title: A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs
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.  more » « less
Award ID(s):
1839317
PAR ID:
10568724
Author(s) / Creator(s):
;
Editor(s):
Mestre, Julián; Wirth, Anthony
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
322
ISSN:
1868-8969
ISBN:
978-3-95977-354-6
Page Range / eLocation ID:
322-322
Subject(s) / Keyword(s):
Homomorphism counting Bounded degeneracy graphs Fine-grained complexity Orbit counting Subgraph counting Mathematics of computing → Graph algorithms Theory of computation → Graph algorithms analysis
Format(s):
Medium: X Size: 19 pages; 843008 bytes Other: application/pdf
Size(s):
19 pages 843008 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
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