Discrete Temporal Constraint Satisfaction Problems

Abstract

Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterises Datalog already for the CSPs of first-order reducts of (Q; <); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be solved in Datalog, and that can be solved in fixed point logic (with or without counting); the classification shows that many of the equivalent conditions in the finite fail to capture expressibility in any of these formalisms already for temporal CSPs.

Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

“…We are going to provide more detailed versions of Theorems 1 and 2, which describe in particular the delineation between the tractable and the NP-complete cases algebraically, in Sections 5 and 8. We would like to emphasize that our proof does not assume or use the dichotomy for CSPs of finite structures, as opposed to some other dichotomy results for CSPs of infinite structures such as [BMM17].…”

“…When the domain is infinite, the complexity of the CSP can be outside NP, and even undecidable [BN06]. But for natural classes of such CSPs there is often the potential for structured classifications, and this has proved to be the case for structures first-order definable over the order (Q, <) of the rationals [BK09] or over the integers with successor [BMM17]. Another classification of this type has been obtained for CSPs where the constraint language is first-order definable over the random (Rado) graph [BP15a], making use of structural Ramsey theory.…”

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

“…Therefore, only by focussing on special classes of infinite-domain CSPs (and VCSPs) is it possible to obtain general complexity results. There is a rich literature on the computational complexity of special classes of infinite-domain CSPs, e.g., [9,8,13,32,7,11,12,5].…”

CLAP: A New Algorithm for Promise CSPs

“…However, while certain infinite-domain CSPs are amenable to algebraic methods, the complexity of infinite-domain CSPs is far from understood, cf. [13,17,18] for recent work.…”