A Complexity Dichotomy for Temporal Valued Constraint Satisfaction Problems

TL;DR

This paper establishes a complete complexity classification for a class of infinite-domain valued constraint satisfaction problems called temporal VCSPs, showing they are either solvable in polynomial time or NP-complete.

Contribution

It provides the first dichotomy theorem for VCSPs over infinite domains, specifically for structures preserved by all order-preserving bijections, using fractional polymorphisms.

Findings

Temporal VCSPs are either in P or NP-complete.

The analysis introduces fractional polymorphisms as a key tool.

This is the first complete dichotomy for infinite-domain VCSPs.

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.