Reasoning about Cardinal Directions between Extended Objects: The Hardness Result

TL;DR

This paper investigates the computational complexity of reasoning with the cardinal direction calculus (CDC), revealing that even incomplete networks are NP-hard to check for consistency, thus clarifying the boundary between tractable and intractable cases.

Contribution

It establishes that allowing unspecified constraints in CDC networks makes the consistency checking problem NP-hard, sharpening the understanding of CDC's computational boundaries.

Findings

Consistency checking of incomplete CDC networks is NP-hard.

The boundary between tractable and intractable CDC subclasses is precisely characterized.

Reduction from 3-SAT demonstrates the intractability result.

Abstract

The cardinal direction calculus (CDC) proposed by Goyal and Egenhofer is a very expressive qualitative calculus for directional information of extended objects. Early work has shown that consistency checking of complete networks of basic CDC constraints is tractable while reasoning with the CDC in general is NP-hard. This paper shows, however, if allowing some constraints unspecified, then consistency checking of possibly incomplete networks of basic CDC constraints is already intractable. This draws a sharp boundary between the tractable and intractable subclasses of the CDC. The result is achieved by a reduction from the well-known 3-SAT problem.