Distance Constraint Satisfaction Problems

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.

💡 Research Summary

The paper investigates the computational complexity of constraint satisfaction problems (CSPs) whose templates Γ are first‑order definable over the structure (ℤ; succ), where succ denotes the successor relation on the integers. The authors focus on the case where Γ is locally finite, meaning that the Gaifman graph of Γ has bounded degree, which restricts the interaction between variables and makes algorithmic analysis feasible despite the infinite domain.

Assuming the widely accepted tractability conjecture for finite‑domain CSPs—specifically the Bulatov‑Jeavons‑Krokhin (BJK) conjecture in its special form for transitive finite templates—the paper delivers a complete trichotomy for locally finite Γ. The three mutually exclusive outcomes are:

1. Polynomial‑time tractability – Γ is homomorphically equivalent to a structure that admits a d‑modular maximum or minimum polymorphism. Such polymorphisms impose a strong symmetry: the operation selects the maximum (or minimum) of its arguments after reducing each argument modulo a fixed integer d. The authors show how any instance of CSP(Γ) can be transformed, using only a bounded number of succ‑applications, into a problem that respects this modular symmetry. Consequently, standard algorithmic techniques (linear programming, dynamic programming, or specialized global‑minimum algorithms) solve the instance in polynomial time.

2. Equivalence to a finite transitive structure – Γ is homomorphically equivalent to a finite structure whose automorphism group acts transitively on its domain. In this situation the infinite‑domain problem collapses to a finite‑domain CSP, and the BJK conjecture directly predicts its complexity (either tractable or NP‑complete). The paper treats this case as a separate branch of the classification, noting that the reduction to a finite template preserves the locally finite property.

3. NP‑completeness – If Γ does not admit a d‑modular max/min polymorphism and is not equivalent to a finite transitive structure, the authors construct polynomial‑time reductions from classic NP‑complete problems such as 3‑SAT or graph 3‑coloring to CSP(Γ). The reductions are carefully designed to respect the bounded degree of the Gaifman graph, ensuring that no unbounded fan‑out is introduced. This demonstrates that any remaining locally finite template yields an NP‑complete CSP.

The technical heart of the work lies in two novel contributions. First, the authors develop a method to extract modular polymorphisms from the successor relation, showing that any locally finite Γ that respects a certain congruence condition can be equipped with a d‑modular max/min operation. Second, they provide a generic reduction framework that, given a template lacking such polymorphisms, embeds arbitrary Boolean formulas into Γ‑instances while keeping the interaction graph sparse.

The overall result is a clean trichotomy theorem: under the BJK conjecture, every locally finite template definable in (ℤ; succ) falls into exactly one of the three categories above. This constitutes the first global complexity classification for an infinite‑domain CSP based solely on the successor relation, extending the finite‑domain dichotomy paradigm to a natural class of arithmetic structures. The paper also discusses how the techniques might be adapted to other infinite domains (e.g., (ℤ; <) or (ℚ; +)) and outlines open problems concerning the removal of the BJK assumption.