Distance Constraint Satisfaction Problems

TL;DR

This paper classifies the computational complexity of certain distance constraint satisfaction problems over the integers, assuming a key conjecture, and identifies conditions for polynomial-time solvability or NP-completeness.

Contribution

It provides a full classification of the complexity of these problems for locally finite templates under a widely held conjecture, extending the understanding of CSPs over infinite domains.

Findings

CSPs are polynomial-time solvable if Gamma has a d-modular polymorphism.

CSPs are NP-complete if Gamma is homomorphically equivalent to a finite transitive structure.

The classification depends on the structure's homomorphic properties and polymorphisms.

Abstract

We study the complexity of constraint satisfaction problems for templates $\Gamma$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), we provide a full classification for the case that Gamma is locally finite (i.e., the Gaifman graph of $\Gamma$ has finite degree). We show that one of the following is true: The structure Gamma is homomorphically equivalent to a structure with a d-modular maximum or minimum polymorphism and $\mathrm{CSP}(\Gamma)$ can be solved in polynomial time, or $\Gamma$ is homomorphically equivalent to a finite transitive structure, or $\mathrm{CSP}(\Gamma)$ is NP-complete.