Distance constraint satisfaction problems

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