Distance Constraint Satisfaction Problems

Abstract

We study the complexity of constraint satisfaction problems for templates $\it\Gamma$ that are first-order definable in $({\mathbb Z}; {\it suc})$, 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 $\it\Gamma$ is locally finite (i.e., the Gaifman graph of $\it\Gamma$ has finite degree). We show that one of the following is true: The structure $\it\Gamma$ is homomorphically equivalent to a structure with a certain majority polymorphism (which we call modular median) and CSP$(\it\Gamma)$ can be solved in polynomial time, or $\it\Gamma$ is homomorphically equivalent to a finite transitive structure, or CSP$(\it\Gamma)$ is NP-complete.