A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations

TL;DR

This paper introduces a new theoretical framework to simplify the proof of the CSP Dichotomy Conjecture and characterizes tractability of CSPs via symmetric polymorphisms, advancing understanding of constraint satisfaction problems.

Contribution

It provides a new proof of the CSP Dichotomy Conjecture using a novel theory of strong subalgebras and linear congruences, and characterizes tractability through XY-symmetric operations.

Findings

New proof of the CSP Dichotomy Conjecture.

Characterization of tractability via infinitely many XY-symmetric polymorphisms.

Development of a theory of strong subalgebras and linear congruences.

Abstract

We develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $\Gamma$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $\Gamma$ that are symmetric on all two-element sets.