Distance Constraint Satisfaction Problems

We study the complexity of constraint satisfaction problems for templates $Γ$ 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 $Γ$ 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 $\mathrm{CSP}(Γ)$ can be solved in polynomial time, or $Γ$ is homomorphically equivalent to a finite transitive structure, or $\mathrm{CSP}(Γ)$ is NP-complete.

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