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）复杂度与算法研究中涉及的代数理论基础。我们统一、简化并拓展了用于研究有限模板CSP的三种方法——用于刻画局部相容性方法可解CSP的吸收理论（JACM'14），以及分别用于独立证明CSP二分律定理的Bulatov理论与Zhuk理论（FOCS'17, JACM'20）。作为第一项贡献，我们提出了关于原始正可定义性的基础定理，并以此推论出Bulatov与Zhuk证明的出发点。作为第二项贡献，我们提出并系统开展了极小泰勒代数的研究。这类代数具有足够广泛的覆盖性，足以将CSP二分律定理的验证限定于该类代数之上，同时展现出异常良好的行为特性。特别地，三种方法中的众多概念在该类代数中趋于一致，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导向一个更简单、更自然的二分律定理证明，该证明将采用更简洁高效的算法，并有助于攻克其他CSP相关问题中的复杂度难题。