Ordering Spatio-Temporal Sequences to Meet Transition Constraints: Complexity and Framework

Time and space are fundamental concepts of study in Artificial Intelligence and, in particular, Knowledge Representation. In this paper, we investigate the task of ordering a temporal sequence of qualitative spatial configurations to meet certain transition constraints. This ordering is constrained by the use of conceptual neighbourhood graphs defined on qualitative spatial constraint languages. In particular, we show that the problem of ordering a sequence of qualitative spatial configurations to meet such transition constraints is \(\mathcal{NP}\)-complete for the the well known languages of RCC-8, Interval Algebra, and Block Algebra. Based on this result, we also propose a framework where the temporal aspect of a sequence of qualitative spatial configurations is constrained by a Point Algebra network, and again show that the enhanced problem is in \(\mathcal{NP}\) when considering the aforementioned languages. Our results lie within the area of Graph Traversal and allow for many practical and diverse applications, such as identifying optimal routes in mobile robot navigation, modelling changes of topology in biological processes, and computing sequences of segmentation steps used in image processing algorithms.