Distance Constraint Satisfaction Problems

We study the complexity of constraint satisfaction problems for templates Γ that are firstorder definable in (Z; succ), the integers with the successor relation. In the case that Γ is locally finite (i.e., the Gaifman graph of Γ has finite degree), we show that Γ is homomorphically equivalent to a structure with a certain majority polymorphism (which we call modular median) and the CSP for Γ can be solved in polynomial time, or Γ is homomorphically equivalent to a finite transitive structure, or the CSP for Γ is NP-complete. 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), this proves that those CSPs have a complexity dichotomy, that is, are either in P or NP-complete.