Searching for an algebra on CSP solutions

Abstract

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism ${\mathbf R}\to {\bf \Gamma}$ between two relational structures, where ${\mathbf R}$ is defined over a domain $V$ and ${\bf \Gamma}$ is defined over a domain $D$. In a fixed template CSP, denoted $CSP({\bf \Gamma})$, the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\mathbf R}$ is unconstrained. We consider the following problem: given a prespecified finite set of algebras ${\mathcal B}$ whose domain is $D$, is it possible to present the solutions set of a given instance of $CSP({\bf \Gamma})$ (which is an input to the problem) as a subalgebra of ${\mathbb A}_1\times ... \times {\mathbb A}_{|V|}$ where ${\mathbb A}_i\in {\mathcal B}$? We study this problem and show that it can be reformulated as an instance of a certain fixed-template CSP, over another template ${\bf \Gamma}^{\mathcal B}$. First, we demonstrate examples of ${\mathcal B}$ for which $CSP({\bf \Gamma}^{\mathcal B})$ is tractable for any, possibly NP-hard, ${\bf \Gamma}$. Under natural assumptions on ${\mathcal B}$, we prove that $CSP({\bf \Gamma}^{\mathcal B})$ can be reduced to a certain fragment of $CSP({\bf \Gamma})$. We also study the conditions under which $CSP({\bf \Gamma})$ can be reduced to $CSP({\bf \Gamma}^{\mathcal B})$. Since the complexity of $CSP({\bf \Gamma}^{\mathcal B})$ is defined by $Pol({\bf \Gamma}^{\mathcal B})$, we study the relationship between $Pol({\bf \Gamma})$ and $Pol({\bf \Gamma}^{\mathcal B})$. It turns out that if $\mathcal{B}$ is preserved by $p\in Pol({\bf \Gamma})$, then $p$ can be extended to a polymorphism of ${\bf \Gamma}^{\mathcal B}$. In the end to demonstrate usefulness of our definitions we study one case when ${\bf \Gamma}$ is not of bounded width, but ${\bf \Gamma}^{\mathcal B}$ is of bounded width (i.e. has a richer structure of polymorphisms).Comment: 34 page