Searching for an algebra on CSP solutions
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The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism R→Γ between two relational structures, where R is defined over a domain V and Γ is defined over a domain D. In fixed template CSPs, denoted CSP(Γ), the right side structure Γ is fixed and the left side structure R is unconstrained. In the last 2 decades it was impressively revealed that those templates Γ are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from R to Γ are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set. This view makes of interest the following formulation: given a prespecified finite set of algebras B whose domain is D, is it possible to present the solutions set of a CSP instance as a subalgebra of A_1× ... × A_|V| where A_i∈ B? We study this formulation and prove that the latter problem itself is an instance of a certain fixed-template CSP, over another template Γ^ B. We prove that CSP(Γ^ B) can be reduced to a certain fragment of CSP(Γ), under natural assumptions on B. We also study the conditions under which CSP(Γ) can be reduced to CSP(Γ^ B). Since the complexity of CSP(Γ^ B) is defined by Pol(Γ^ B), we study the relationship between Pol(Γ) and Pol(Γ^ B). It turns out that if B is preserved by some p∈ Pol(Γ), then p can be extended to a polymorphism of Γ^ B.
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