A Complexity Dichotomy for Temporal Valued Constraint Satisfaction Problems

We study the complexity of the valued constraint satisfaction problem (VCSP) for every valued structure with the domain ${\mathbb Q}$ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to the (classical) constraint satisfaction problem: a relational structure is preserved by all order-preserving bijections if and only if all its relations have a first-order definition in $({\mathbb Q};<)$, and the CSPs for such structures are called temporal CSPs. Many optimization problems that have been studied intensively in the literature can be phrased as a temporal VCSP. We prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the concept of fractional polymorphisms. This is the first dichotomy result for VCSPs over infinite domains which is complete in the sense that it treats all valued structures with a given automorphism group.

💡 Research Summary

The paper establishes the first complete complexity dichotomy for valued constraint satisfaction problems (VCSPs) over an infinite domain, specifically the rational numbers ℚ, when the template is “temporal” – that is, it is invariant under all order‑preserving bijections of ℚ (the automorphism group Aut(ℚ;<)). Temporal VCSPs generalize the well‑studied temporal CSPs, but now allow arbitrary finite or infinite cost values rather than just {0,∞}.

Background. VCSPs extend classical CSPs by assigning a cost to each constraint; the goal is to find an assignment whose total cost does not exceed a given threshold. For finite‑domain templates, a celebrated dichotomy theorem (Thapper‑Živný, Kolmogorov‑Krokhin‑Rolínek, Bulatov, Zhuk) shows that every VCSP is either solvable in polynomial time (P) or NP‑complete, with the boundary characterized by the existence of certain fractional polymorphisms. Over infinite domains, however, a general classification is impossible; only specific automorphism groups (e.g., the full symmetric group, or Aut(ℚ;<) for temporal CSPs) have yielded dichotomies for the decision version (CSP).

Main contribution. The authors prove that for every valued structure A with domain ℚ that is preserved by all order‑preserving bijections, the problem VCSP(A) falls into exactly one of two categories:

1. Tractable case (P). A admits a fractional polymorphism consisting of a single operation with probability 1. In practice this operation is one of the basic monotone functions on ℚ (e.g., min, max, or averaging). The presence of such a polymorphism guarantees that the standard linear‑programming (LP) relaxation of the VCSP is exact: the LP optimum equals the true optimum, and a simple rounding procedure (which respects the order) yields an optimal integral solution. Consequently, VCSP(A) can be solved in polynomial time.

2. Hard case (NP‑complete). If A does not have the above fractional polymorphism, then there exists at least one cost function in A that is pp‑expressible (i.e., definable by a primitive positive formula) and whose structure distinguishes different Aut(ℚ;<)‑orbits. Using generalized pp‑constructions, the authors show that such a function can be reduced from known NP‑hard temporal VCSPs (e.g., minimum feedback arc set, directed multicut, Steiner multicut). Hence VCSP(A) is NP‑complete.

The dichotomy is complete: every temporal template falls into exactly one of the two cases, and no intermediate complexity is possible. The proof proceeds in stages. First, the authors treat the extreme case where the automorphism group is the full symmetric group (all permutations), which serves as a warm‑up and supplies technical lemmas. Then they extend the arguments to the full Aut(ℚ;<) setting, exploiting the oligomorphic nature of this group (finitely many orbits of k‑tuples for each k) to translate valued relations into finite families of first‑order formulas over (ℚ;<). This translation makes it possible to reason about pp‑expressibility and to construct the required reductions.

Technical insights.

• Fractional polymorphisms: Unlike the finite‑domain situation where a rich convex combination of operations may be needed, the temporal setting forces any tractable polymorphism to be deterministic (probability 1). This simplification stems from the high symmetry of ℚ under order‑preserving bijections.
• LP exactness: The authors prove that the basic LP relaxation, which includes variables for each tuple of the domain, is always tight when the deterministic polymorphism exists. The proof uses the fact that the polymorphism preserves the order and therefore the feasible region of the LP is closed under the operation.
• pp‑expressibility: For hardness, the key is to identify a cost function whose feasibility relation partitions ℚᵏ into finitely many Aut(ℚ;<)‑orbits, each definable by a quantifier‑free formula. Such a function can simulate the cost structure of known NP‑hard temporal VCSPs. The reduction leverages generalized pp‑constructions that allow the composition of primitive positive definitions with auxiliary variables.
• Oligomorphic automorphism groups: The paper relies heavily on the fact that Aut(ℚ;<) is oligomorphic, guaranteeing that any valued relation takes only finitely many distinct values. This finiteness is crucial for both the algebraic characterization and the algorithmic treatment.

Implications and applications. The dichotomy immediately classifies a wide range of optimization problems that can be expressed as temporal VCSPs, including directed feedback arc set, various multicut variants, Steiner multicut, and many “min‑CSP” problems studied in database theory and phylogenetics. For each such problem, the paper’s criteria tell us whether a polynomial‑time algorithm exists (typically via LP) or whether the problem is NP‑hard. Moreover, the result provides a template for extending dichotomies to other infinite‑domain VCSPs with well‑structured automorphism groups.

Relation to prior work. The result generalizes the temporal CSP dichotomy of Bodirsky and Kára and the finite‑domain VCSP dichotomy of Thapper‑Živný and others. It also confirms Conjecture 9.3 from