Asking the metaquestions in constraint tractability

TL;DR

This paper investigates the complexity of determining whether certain algebraic properties (polymorphisms) exist in structures related to constraint satisfaction problems, revealing NP-completeness results for key tractability conditions.

Contribution

It systematically analyzes the complexity of deciding polymorphism existence in structures, proving NP-completeness for conditions linked to CSP tractability and bounded width.

Findings

Deciding if a structure admits a polymorphism from certain classes is NP-complete.

NP-completeness of deciding CSP(H) bounded width.

Results connect algebraic properties to computational complexity in CSPs.

Abstract

The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of the CSP is as the problem of deciding, given a pair (G,H) of relational structures, whether or not there is a homomorphism from the first structure to the second structure. The CSP is in general NP-hard; a common way to restrict this problem is to fix the second structure H, so that each structure H gives rise to a problem CSP(H). The problem family CSP(H) has been studied using an algebraic approach, which links the algorithmic and complexity properties of each problem CSP(H) to a set of operations, the so-called polymorphisms of H. Certain types of polymorphisms are known to imply the polynomial-time tractability of $CSP(H)$, and others are conjectured to do so. This article systematically studies---for various classes of polymorphisms---the computational complexity of deciding whether or not a given structure H admits a polymorphism from the class. Among other results, we prove the NP-completeness of deciding a condition conjectured to characterize the tractable problems CSP(H), as well as the NP-completeness of deciding if CSP(H) has bounded width.