A Complexity Dichotomy in Spatial Reasoning via Ramsey Theory

TL;DR

This paper establishes a complexity dichotomy for spatial reasoning CSPs using Ramsey theory, showing conditions for polynomial-time solvability and classifying expressibility in Datalog.

Contribution

It introduces a new approach leveraging Ramsey theory to determine complexity classifications for CSPs in spatial reasoning, specifically for RCC5 expansions.

Findings

RCC5 has a Ramsey order expansion.

Conditions identified for applying polynomial-time reduction methods.

Classified which CSPs are expressible in Datalog.

Abstract

Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k-types, for a sufficiently large k. We give sufficient conditions when this method can be applied and illustrate how to use the general results to prove a new complexity dichotomy for first-order expansions of the basic relations of the well-studied spatial reasoning formalism RCC5. We also classify which of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory; we prove that RCC5 has a Ramsey order expansion.