Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

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

The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

We prove that an $\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\alpha$, $\beta$, $s$ satisfying the identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page

Algebraic Approach to Constraint Satisfaction Problems

A constraint satisfaction problem (CSP) asks for an assignment of appropriate values to variables subject to a given set of constraints. Examples of CSPs are found in virtually every technical discipline. We would like to discover methods for deciding when a CSP is “easy,” or tractable and when it is “hard” or intractable. Theories from universal algebra have turned out to be applicable in obtaining results on the tractability of CSPs. Specifically, each CSP can be converted to an algebra. The so-called absorption property of an algebra plays a key role in proofs related to the complexity of the corresponding CSPs. We developed some methods that enabled us to classify those absorption-free 4-element members of a certain class of algebras, called commutative idempotent binars (CIBs), which are not known to be tractable. Moreover, we determined the absorbing subalgebras of each 4-element CIB that is not absorption-free. Previously, few examples of either absorption-free CIBs or CIBs with absorbing subalgebras were known. This recent progress has therefore added to the general knowledge of the tractability of algebras and their corresponding CSPs

The number of clones determined by disjunctions of unary relations

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $\Gamma$. We prove that whenever $\Gamma$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $\Gamma$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $\Gamma$.Comment: manuscript to be published in Theory of Computing System