Temporal Flexibility Revisited: Maximizing Flexibility by Computing Bipartite Matchings

Abstract

We discuss two flexibility metrics for Simple Temporal Networks (STNs): the so-called naive flexibility metric based on the difference between earliest and latest starting times of temporal variables, and a recently proposed concurrent flexibility metric. We establish an interesting connection between the computation of these flexibility metrics and properties of the minimal distance matrix D_S of an STN S: the concurrent flexibility metric can be computed by finding a minimum weight matching of a weighted bipartite graph completely specified by D_S, while the naive flexibility metric corresponds to computing a maximum weight matching in the same graph. From a practical point of view this correspondence offers an advantage: instead of using an O(n⁵) LP-based approach, reducing the problem to a matching problem we derive an O(n³) algorithm for computing the concurrent flexibility metric.