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.

翻译：我们研究了定义在有理数域${\mathbb Q}$上且被所有保序双射所保持的每个值结构的值约束满足问题（VCSP）的复杂性。这类VCSP被称为时态VCSP，类比于（经典）约束满足问题：一个关系结构被所有保序双射所保持，当且仅当其所有关系在$({\mathbb Q};<)$中具有一阶定义，此类结构的CSP被称为时态CSP。文献中深入研究的许多优化问题都可以表述为时态VCSP。我们证明时态VCSP要么属于P类，要么是NP完全的。我们的分析使用了分数多态的概念。这是首个关于无限域上VCSP的完整二分法结果，其完整性体现在它处理了具有给定自同构群的所有值结构。