Modular Counting over 3-Element and Conservative Domains
Andrei A. Bulatov, Amirhossein Kazeminia
TL;DR
The paper studies modular counting CSPs, counting homomorphisms to a fixed relational structure ${\mathcal{H}}$ modulo a prime $p$, and introduces $p$-automorphic polynomials to reduce domain sizes via ${\mathcal{H}}^{f}$. It develops a comprehensive toolkit—including automorphism-stable sets, Möbius inversion, indicator problems, and $p$-indicator obstructions—to derive tractability or hardness for two target classes: $3$-element structures and $p$-conservative structures, with strong results when Mal'tsev polymorphisms exist in the modular sense. A central theme is that either a Mal'tsev polymorphism in $ig\langle {\mathcal{H}} \big\rangle_p$ (or in $ig\langle {\mathcal{H}}^{f} \big\rangle_p$ for automorphic reductions) yields polynomial-time solvability, or the presence of $p$-indicator obstructions drives ${\#_pP}$-hardness. The work generalizes modular counting beyond graphs to general relational structures, offering a principled pathway to dichotomies via domain-reducing automorphisms and modular definability concepts, and clarifies the role of $p$-subalgebras and partition-function complexity in this setting.
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.