Constraint Satisfaction Tractability from Semi-lattice Operations on Infinite Sets

TL;DR

This paper extends a fundamental finite-domain CSP tractability result to certain infinite-domain CSPs with well-behaved automorphism groups, showing they remain polynomial-time solvable under semi-lattice preservation.

Contribution

It generalizes Jeavons et al.'s theorem to infinite domains with controlled automorphism orbit growth, broadening the class of tractable CSPs.

Findings

Infinite domain CSPs with sub-exponential orbit growth are polynomial-time solvable.

Preservation under semi-lattice operations implies tractability for these CSPs.

Many such CSPs are not solvable by Datalog, unlike finite cases.

Abstract

A famous result by Jeavons, Cohen, and Gyssens shows that every constraint satisfaction problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal-algebraic approach to a systematic theory of tractability and hardness in finite domain constraint satisfaction. Not surprisingly, the theorem of Jeavons et al. fails for arbitrary infinite domain CSPs. Many CSPs of practical interest, though, and in particular those CSPs that are motivated by qualitative reasoning calculi from Artificial Intelligence, can be formulated with constraint languages that are rather well-behaved from a model-theoretic point of view. In particular, the automorphism group of these constraint languages tends to be large in the sense that the number of orbits of n-subsets of the automorphism group is bounded by some function in n. In this paper we present a generalization of the theorem by Jeavons et al. to infinite domain CSPs where the number of orbits of n-subsets grows sub-exponentially in n, and prove that preservation under a semi-lattice operation for such CSPs implies polynomial-time tractability. Unlike the result of Jeavons et al., this includes many CSPs that cannot be solved by Datalog.