Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

TL;DR

This paper introduces new characterizations of Taylor varieties using absorbing subalgebras and cyclic terms, providing elementary proofs for key conjectures in the algebraic complexity of the Constraint Satisfaction Problem.

Contribution

It offers two novel characterizations of finitely generated Taylor varieties, simplifying the understanding of their structure and implications for the CSP complexity dichotomy.

Findings

Reproves the algebraic dichotomy conjecture for CSPs using new characterizations.

Provides elementary proofs for existing conjectures and characterizations.

Establishes connections between Taylor varieties, absorbing subalgebras, and cyclic terms.

Abstract

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymorphisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and self-contained way.