Asking the Metaquestions in Constraint Tractability

Author:

Chen Hubie1,Larose Benoit2

Affiliation:

1. Universidad del País Vasco and IKERBASQUE, Basque Foundation for Science, Spain

2. Champlain Regional College and Université du Québec a Montréal

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 generally 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.

Funder

Spanish Project MINECO COMMAS

Basque

FRQNT and NSERC

Basque Project

Publisher

Association for Computing Machinery (ACM)

Subject

Computational Theory and Mathematics,Theoretical Computer Science

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