Finding K shortest and dissimilar paths

Abstract

The purpose of the K dissimilar paths problem is to find a set of K paths, between the same pair of nodes, which share few arcs. The problem has been addressed from an application point of view, and integer programming formulations have also been introduced recently. In the present work, it is assumed that each arc is assigned with a cost, and the goal is then to find K dissimilar paths while simultaneously minimizing the total cost. Some of the previous formulations: one minimizing the number of repeated arcs, another one minimizing the number of arc repetitions, as well as modifications that bound the number of paths in which the arcs appear, are extended with a cost function. Properties of the resulting biobjective problems are studied and the ϵ-constraint method is adapted to solve them using a decreasing and an increasing strategy for updating ϵ. These methods are tested for finding sets of 10 paths in random and grid instances to assess the efficiency of the ϵ-constraint methods and the performance of the formulations to calculate shortest and dissimilar paths. Results show that minimizing the number of arc repetitions produces efficient solutions with higher dissimilarities faster than minimizing the number of repeated arcs. The cost range of the solutions is similar in both approaches. Additionally, bounding the number of paths in which each arc appears improves the quality of the solutions as to the dissimilarity while worsening its cost.