Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras

TL;DR

This paper unifies and extends algebraic approaches to the CSP, introduces minimal Taylor algebras, and aims to simplify the proof of the CSP Dichotomy Theorem and related complexity questions.

Contribution

It presents an elementary theorem on primitive positive definability and initiates the systematic study of minimal Taylor algebras, unifying key algebraic approaches to CSP.

Findings

Many concepts from three approaches coincide in minimal Taylor algebras

The class of minimal Taylor algebras suffices for CSP Dichotomy proof

The approach simplifies understanding of CSP complexity

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.