On the induced problem for fixed-template CSPs

TL;DR

This paper explores the algebraic structure of fixed-template CSPs, showing how solution sets can be represented as subalgebras and establishing conditions for reductions between related CSP problems.

Contribution

It introduces a reformulation of the solution set problem as a fixed-template CSP over a new template and analyzes reductions based on algebraic properties.

Findings

CSP solution sets can be expressed as subalgebras of product structures.

If the algebra set is tractable, the related CSP with input prototype is also tractable.

Reductions between CSPs depend on primitive positive definability of algebra relations.

Abstract

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 $\mathbf{\Gamma}$.