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.
Highlights
The “CSP-dichotomy conjecture” of Tomas Feder and Moshe Vardi [FV99] asserts that every constraint satisfaction problem (CSP) over a fixed finite constraint language is either NP-complete or tractable. (The current status of the conjecture is discussed in the note at the end of this introduction.)A discovery of Jeavons, Cohen and Gyssens in [JCG97]—later refined by Bulatov, Jeavons and Krokhin in [BJK05]—was the ability to transfer the question of the complexity of the CSP over a set of relations to a question of algebra
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
Summary
The “CSP-dichotomy conjecture” of Tomas Feder and Moshe Vardi [FV99] asserts that every constraint satisfaction problem (CSP) over a fixed finite constraint language is either NP-complete or tractable. (The current status of the conjecture is discussed in the note at the end of this introduction.). A discovery of Jeavons, Cohen and Gyssens in [JCG97]—later refined by Bulatov, Jeavons and Krokhin in [BJK05]—was the ability to transfer the question of the complexity of the CSP over a set of relations to a question of algebra These authors showed that the complexity of any particular CSP depends solely on the polymorphisms of the constraint relations, that is, the functions preserving all the constraints. Key words and phrases: constraint satisfaction problem, universal algebra, dichotomy conjecture, Mal’tsev condition, Taylor term, absorbing subalgebra.
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