Universal Algebraic Methods for Constraint Satisfaction Problems

TL;DR

This paper introduces new algebraic methods to analyze the complexity of constraint satisfaction problems, especially focusing on small commutative idempotent binars, and proves their tractability.

Contribution

It develops novel techniques for classifying small algebras' CSPs and proves that all CIBs of size up to 4 have tractable CSPs.

Findings

All CIBs of size at most 4 have tractable CSPs.

New algebraic methods for analyzing small algebras.

Advances towards classifying CSP complexity for specific algebra classes.

Abstract

After substantial progress over the last 15 years, the "algebraic CSP-dichotomy conjecture" reduces to the following: every local constraint satisfaction problem (CSP) associated with a finite idempotent algebra is tractable if and only if the algebra has a Taylor term operation. Despite the tremendous achievements in this area (including recently announce proofs of the general conjecture), there remain examples of small algebras with just a single binary operation whose CSP resists direct classification as either tractable or NP-complete using known methods. In this paper we present some new methods for approaching such problems, with particular focus on those techniques that help us attack the class of finite algebras known as "commutative idempotent binars" (CIBs). We demonstrate the utility of these methods by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.