An algebraic approach to Borel CSPs

TL;DR

This paper applies algebraic methods to Borel CSPs, establishing their complexity classification and linking algebraic properties to descriptive set theoretic complexity, including completeness results and characterizations.

Contribution

It introduces an algebraic framework for analyzing Borel CSPs, connecting algebraic properties with descriptive set theoretic complexity classifications and providing new characterizations.

Findings

Non-Taylor structures lead to $oldsymbol{ ext{Σ}}^1_2$-complete Borel CSPs

NP-complete CSPs correspond to $oldsymbol{ ext{Σ}}^1_2$-complete Borel CSPs under the CSP Dichotomy

Structures with all solvable Borel instances having Borel solutions are exactly width 1 structures

Abstract

We adapt tools from the algebraic approach to constraint satisfaction problems to answer descriptive set theoretic questions about Borel CSPs. We show that if a structure $\mathcal D$ does not have a Taylor polymorphism, then the corresponding Borel CSP is $\mathbf{\Sigma}^1_2$-complete. In particular, by the CSP Dichotomy Theorem, if $\operatorname{CSP}(\mathcal D)$ is $\mathrm{NP}$-complete, then the Borel version, $\operatorname{csp}_B(\mathcal D)$, is $\mathbf{\Sigma}^1_2$-complete (assuming $\mathrm{P}\not=\mathrm{NP}$). We also have partial converses, such as a descriptive analogue of the Hell--Ne\v set\v ril theorem characterizing $\mathbf{\Sigma}^1_2$-complete graph homomorphism problems. We show that the structures where every solvable Borel instance of their CSP has a Borel solution are exactly the width 1 structures. And, we prove a handful of results bounding the projective complexity of certain bounded width structures.