Eecient Algorithms for Networks of Quantitative Temporal Constraints

This paper describes two algorithms for networks of quantitative temporal constraints. The rst one is an incremental algorithm for checking the consistency of a set of constraints: it is based on the directional path consistency property, and it is a complete algorithm. The second algorithm relies on a property of shortest paths in a directional path consistent network. We prove this property and present an eecient linear-time algorithm that has a lower cost than usual shortest-path algorithm such as Bellman-Ford algorithm. These two algorithms form a complete system for managing quantitative temporal relationships. 1 Motivations Temporal constraints systems are widely used for temporal reasoning: these systems allow to deal ef-ciently with temporal relationships describing the relative positions of some temporal entities, such as instants or intervals. Given a set of such relationships , we are mainly interested in checking the satis-ability, or the consistency, of this set of constraints, and in computing the minimal relation between any two temporal entities. Consistency means that there exists a solution to the set of constraints. Depending on the formalism, point based or interval based, a solution is an assignment of a point or an interval on a time line (e.g. the set of rational numbers) to every temporal entity, according to its sort, and such that all temporal constraints are satissed by this assignment. In this article we shall restrain to point based temporal relations, i.e. relations describing the order between any two instants. Such temporal constraints are interesting because consistency can be decided in polynomial time 7]. An important requirement for the practical use of a consistecy checking algorithm is its incrementality, for example, when using a CLP-like resolution principle 1]. This both means that such an algorithm can be used to add one constraint at a time, retaining all previously encountered constraints as an internal statee, and that the cost of adding one constraint to this structure is less than the straightforward approach of checking again the consistency in one operation with the set of all constraints. In section 3 we propose an incremental algorithm to check the consistency of a set of quantitative temporal constraints between points. This algorithm extends some work done in 3]. Contrary to usual algorithms for temporal constraints, our algorithm does not compute the minimal relation between any two points in the network. This justiies the second algorithm that we propose in Section 4: It computes …