This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates - absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM'14), and Bulatov's and Zhuk's theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS'17, JACM'20). As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov's and Zhuk's proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough so that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in the class, which is in striking contrast with the general setting. We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.
翻译:本文侧重于研究限制满意度问题的复杂性和算法(CSP)研究所依据的代数理论。 我们统一、简化和扩展了为研究CSP在有限模板基础上的三种方法(吸收理论(JACM'14)和Bulatov's和Zhuk的理论(用于CSP Dichotommy Theorem (FOCS'17, JACM'20)的两种独立证据)而开发的三种方法(CSP 17, JACM'20)的缩略论 ) 。 我们首先提出了原始正定义定义的基本理论,并用它来获取Bulatov 和 Zhuk 证据的起始点,作为缩略缩模板(JACM'14) 和 Bulatov's 和 Zhuk 理论(Zhuk 理论) 的缩写, 用于描述CSP Digebras 的缩略图和 Zhuk 理论的缩略图。 这种代数非常广泛,足以核实CSP 简化的缩略图(FOC'17), 但表现得异常良好。 。 特别是, 三种方法中的许多概念, 从三种概念, 将最终的缩略图的缩略图中, 我们的缩略图的缩略图的缩图的缩图与C 的缩图的缩图的缩图结果将比。