Tractable Combinations of Temporal CSPs

TL;DR

This paper investigates the computational complexity of combined temporal constraint satisfaction problems, showing they are either efficiently solvable or NP-complete, based on algebraic properties of temporal structures.

Contribution

It characterizes the complexity of CSPs formed by combining two temporal theories, establishing a dichotomy in their computational complexity.

Findings

CSP$(T_1 up T_2)$ is either in P or NP-complete.

The complexity depends on algebraic properties of the structures involved.

A purely algebraic proof relates to the lattice of clones over $Q$.

Abstract

The constraint satisfaction problem (CSP) of a first-order theory T is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of T. We study the computational complexity of CSP$(T_1 \cup T_2)$ where $T_1$ and $T_2$ are theories with disjoint finite relational signatures. We prove that if $T_1$ and $T_2$ are the theories of temporal structures, i.e., structures where all relations have a first-order definition in $(Q;<)$, then CSP$(T_1 \cup T_2)$ is in P or NP-complete. To this end we prove a purely algebraic statement about the structure of the lattice of locally closed clones over the domain $Q$ that contain Aut$(Q;<)$.