A Complexity Dichotomy for Temporal Valued Constraint Satisfaction Problems
Manuel Bodirsky, Édouard Bonnet, Žaneta Semanišinová
TL;DR
The paper resolves a complete P vs NP-complete dichotomy for temporal VCSPs, i.e., VCSPs over ${\mathbb Q}$ preserved by all order-preserving bijections. The authors develop and apply an algebraic framework based on fractional polymorphisms, pp-expressions, and pp-constructions to separate tractable cases (polynomial-time solvable) from hard cases (NP-hard), with tractability achieved when the crisp reduct is preserved by $ll$ or $ll^*$ (or related base operations) and hardness arising from pp-constructing $K_3$ via certain temporal relations. A central contribution is showing that these temporal VCSPs are completely classified by the automorphism group structure and first-order definability in $(\mathbb{Q};<)$, yielding a decidable, sharp boundary between P and NP-hard instances. The results extend the Resilience-VCSPs program to infinite domains and provide a foundation for transferring complexity classifications from temporal CSPs to their optimization counterparts, offering insights for other infinite-domain VCSPs and potential extensions to broader oligomorphic automorphism groups.
Abstract
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.