On the induced problem for fixed-template CSPs

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism $\mathbf{R}\to \mathbf{\Gamma}$ between two relational structures, where $\mathbf{R}$ is defined over a domain $V$ and $\mathbf{\Gamma}$ is defined over a domain $D$. In a fixed template CSP, denoted $\rm{CSP}(\mathbf{\Gamma})$, the right side structure $\mathbf{\Gamma}$ is fixed and the left side structure $\mathbf{R}$ is unconstrained. In the last two decades it was discovered that the reasons that make fixed template CSPs polynomially solvable are of algebraic nature, namely, templates that are tractable should be preserved under certain polymorphisms. From this perspective the following problem looks natural: given a prespecified finite set of algebras ${\mathcal B}$ whose domain is $D$, is it possible to present the solution set of a given instance of $\rm{CSP}(\mathbf{\Gamma})$ 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 $\mathbf{\Gamma}^{\mathcal B}$. We study conditions under which $\rm{CSP}(\mathbf{\Gamma})$ can be reduced to $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$. This issue is connected with the so-called CSP with an input prototype, formulated in the following way: given a homomorphism from $\mathbf{R}$ to $\mathbf{\Gamma}^{\mathcal B}$ find a homomorphism from $\mathbf{R}$ to $\mathbf{\Gamma}$. We prove that if ${\mathcal B}$ contains only tractable algebras, then the latter CSP with an input prototype is tractable. We also prove that $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$ can be reduced to $\rm{CSP}(\mathbf{\Gamma})$ if the set ${\mathcal B}$, treated as a relation over $D$, can be expressed as a primitive positive formula over $\mathbf{\Gamma}$.