The Complexity of Combinations of Qualitative Constraint Satisfaction Problems

TL;DR

This paper investigates the computational complexity of combined qualitative constraint satisfaction problems, showing that for many theories, Nelson-Oppen conditions are both necessary and sufficient for polynomial-time solvability unless P=NP.

Contribution

It extends the understanding of the Nelson-Oppen framework by establishing necessity of their conditions for polynomial-time tractability in a broad class of theories.

Findings

Nelson-Oppen conditions are necessary and sufficient for polynomial-time solvability in many cases.

For a large class of $oldsymbol{}$-categorical theories, the conditions characterize tractability.

The results imply that outside these conditions, the combined CSPs are unlikely to be efficiently solvable.

Abstract

The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of $\mathrm{CSP}(T_1 \cup T_2)$ under the assumption that $\mathrm{CSP}(T_1)$ and $\mathrm{CSP}(T_2)$ are decidable (or polynomial-time decidable). We show that for a large class of $\omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of $\mathrm{CSP}(T_1 \cup T_2)$ (unless P=NP).