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Modular Counting over 3-Element and Conservative Domains

Authors Andrei A. Bulatov, Amirhossein Kazeminia

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Constraint and logic programming
Keywords
  • Constraint Satisfaction Problem
  • Modular Counting

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Abstract

In the Constraint Satisfaction Problem (CSP for short) the goal is to decide the existence of a homomorphism from a given relational structure {G} to a given relational structure {H}. If the structure {H} is fixed and {G} is the only input, the problem is denoted CSP({H}). In its counting version, #CSP({H}), the task is to find the number of such homomorphisms. The CSP and #CSP have been used to model a wide variety of combinatorial problems and have received a tremendous amount of attention from researchers from multiple disciplines.
In this paper we consider the modular version of the counting CSPs, that is, problems of the form #_pCSP({H}) of counting the number of homomorphisms to {H} modulo a fixed prime number p. Modular counting has been intensively studied during the last decade, although mainly in the case of graph homomorphisms. Here we continue the program of systematic research of modular counting of homomorphisms to general relational structures. The main results of the paper include a new way of reducing modular counting problems to smaller domains and a study of the complexity of such problems over 3-element domains and over conservative domains, that is, relational structures that allow to express (in a certain exact way) every possible unary predicate.

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Andrei A. Bulatov and Amirhossein Kazeminia. Modular Counting over 3-Element and Conservative Domains. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.22

Author Details

Andrei A. Bulatov
  • Simon Fraser University, Burnaby, Canada
Amirhossein Kazeminia
  • University of Oxford, UK

Funding

  • Bulatov, Andrei A.: This research is supported by a NSERC Discovery Grant.
  • Kazeminia, Amirhossein: UKRI EP/X024431/1.

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