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：本文聚焦于约束满足问题（CSP）复杂度与算法研究背后的代数理论。我们统一、简化并拓展了针对有限模板CSP研究的三种方法——用于刻画局部一致性方法可解CSP的吸收理论（JACM'14），以及分别独立证明CSP二分定理（FOCS'17, JACM'20）的布拉托夫理论与茹科夫理论。作为第一项贡献，我们提出了关于原始正可定义性的初等定理，并借此将布拉托夫与茹科夫证明的起点作为推论导出。作为第二项贡献，我们提出并系统性地开展了对极小泰勒代数的研究。这类代数具有足够广泛的涵盖性，使得仅在此类代数上验证CSP二分定理便已足够，同时其性质又异常规整。尤为显著的是，三种方法中的诸多概念在此类代数中相互吻合，这与一般情形形成鲜明对比。我们相信，本文开创的理论将最终催生基于更简洁高效算法的二分定理简化证明，并有助于攻克其他CSP相关问题中的复杂性难题。