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.

翻译：在约束满足问题（简称CSP）中，目标在于判定从给定关系结构$G$到给定关系结构$H$是否存在同态映射。若结构$H$固定且$G$为唯一输入，该问题记作$CSP(H)$。在其计数版本$\#CSP(H)$中，任务则是计算此类同态映射的数量。CSP与#CSP已被广泛用于建模各类组合问题，并受到来自多学科研究者的极大关注。本文研究计数CSP的模运算版本，即形如$\#_pCSP(H)$的问题，其目标为计算到$H$的同态映射数量对固定素数$p$取模的结果。模计数在过去十年间得到深入研究，但主要集中于图同态情形。本文延续了对一般关系结构同态模计数的系统性研究计划。论文的主要成果包括：提出一种将模计数问题归约至更小域的新方法，并系统分析了此类问题在三元域及保守域（即允许以某种精确方式表达所有可能一元谓词的关系结构）上的计算复杂度。