Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras
Libor Barto, Zarathustra Brady, Andrei Bulatov, Marcin Kozik, Dmitriy Zhuk
TL;DR
This work unifies three algebraic approaches to the CSP over finite templates—absorption theory, Bulatov’s edge framework, and Zhuk’s four-types theory—by introducing and exploiting minimal Taylor algebras. The authors prove that within minimal Taylor algebras, the three approaches collapse into a coherent structure with simplifying features: 2-absorption coincides with projectivity, 3-absorption aligns with centers, and edges become highly structured, allowing a single witnessing operation to capture all absorptions and edges. Two main contributions emerge: (i) an elementary pp-definability theorem that yields starting points for Bulatov/Zhuk constructions as corollaries, and (ii) a thorough systematic study of minimal Taylor algebras showing strong connections among the three approaches, including a bound on the number of minimal Taylor clones and a path toward a simpler, natural proof of the Dichotomy Theorem. The results suggest that a unified, more natural algorithmic framework for CSP dichotomy is within reach, with implications for broader complexity questions in CSP-related problems and universal algebra.
Abstract
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 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 this 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.