The Complexity of Reasoning about Spatial Congruence

TL;DR

This paper introduces a new algebra for spatial reasoning, proves the NP-completeness of its satisfiability problem, and classifies its tractable subclasses, advancing understanding of computational complexity in spatial reasoning.

Contribution

It presents a novel algebra for spatial congruence, establishes NP-completeness, and classifies maximal tractable subclasses within this algebra.

Findings

NP-completeness of the satisfiability problem in MC-4

Identification of three maximal tractable subclasses

Complete classification of tractability based on relation subsets

Abstract

In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subsets of algebras. The main purpose of these researches is to describe a restricted set of maximal tractable subalgebras, ideally in an exhaustive fashion with respect to the hosting algebras. In this paper we introduce a novel algebra for reasoning about Spatial Congruence, show that the satisfiability problem in the spatial algebra MC-4 is NP-complete, and present a complete classification of tractability in the algebra, based on the individuation of three maximal tractable subclasses, one containing the basic relations. The three algebras are formed by 14, 10 and 9 relations out of 16 which form the full algebra.