Discrete Temporal CSPs et al

Abstract

We revisit the complexity classification of constraint satisfaction problems (CSPs) for first-order reducts of $(\mathbb{Z},<)$, established by Bodirsky, Martin, and Mottet in 2018. Following their approach, we give a self-contained proof of this classification, correcting several errors in the argument. We further show that, with the exception of cases arising from finite-domain CSPs, all tractable cases are solved by a well-known uniform algorithm, eliminating the need for the collection of ad-hoc algorithms used in the original paper. A central component of the proof allows the analysis to be restricted to first-order reducts of $(\mathbb{Z},<)$ that contain the successor relation. We substantially simplify this reduction and generalize the underlying construction. This generalization allows us to apply the same technique to first-order reducts of the triangular lattice graph and of infinite regular trees, yielding a complete complexity classification for these classes of structures, in particular a $\mathsf{P}$ versus $\mathsf{NP}$-complete dichotomy.