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）复杂度与算法研究的代数理论基础。我们统一、简化并扩展了针对有限模板CSP研究所发展的三种方法中的部分内容——用于刻画可由局部一致性方法求解的CSP的吸收理论（JACM'14），以及分别用于CSP二分定理两个独立证明的布拉托夫理论与茹科夫理论（FOCS'17, JACM'20）。作为第一个贡献，我们提出了关于原始正可定义性的基础定理，并将其作为推论导出布拉托夫与茹科夫证明的起点。作为第二个贡献，我们提出并系统开展了最小泰勒代数的研究。这类代数具有足够广泛的覆盖性——仅需在该代数类上验证CSP二分定理即可，同时仍保持优异的数学性质。特别地，三种方法中的诸多概念在该代数类中趋于一致，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为二分定理提供更简洁、更自然的证明（采用更简单高效的算法），并有助于解决其他CSP相关问题的复杂度难题。