Finding K dissimilar paths: Single-commodity and discretized flow formulations

While finding a path between two nodes is the basis for several applications, the need for alternative paths also may have various motivations. For instance, it can be of interest to ensure reliability in a telecommunications network, to reduce the consequences of possible accidents in the transportation of hazardous materials or to decrease the risk of robberies in money distribution. All these applications involve the search for a set of paths that are as dissimilar as possible, with respect to the arcs in them. In this work, we present linear integer programming formulations that intend to find K dissimilar paths. The new models derive from a straightforward single-commodity flow formulation of the K paths problem, to which extra constraints are added to account for the concept of dissimilarity. In particular, in each formulation a different measure of the similarity between the K paths is minimized: the number of repeated arcs, the number of arc reuses and the number of pairwise arc overlaps. The third measure originates a non-linear model that is linearized through a discretized model reformulation technique. These formulations are complemented by a polynomial algorithm that allows to extract K loopless paths from the solutions output by each of them.The formulations are tested for random and grid networks with respect to the solutions' average and minimum dissimilarity and the run time. Two of them stand out in terms of both those measures, when compared to an iterative approach from the literature. The best one regarding the average dissimilarity finds 20 paths with average dissimilarity of 99.5% in about 168 s for random networks of 5 000 nodes and 100 000 arcs. The same approach finds 20 paths for grid instances of size 100 x 100 in less than 18 s. The other formulation requires about 11 s for the same random instances, with a 0.7% deviation from the best known average dissimilarity; while it requires less than 7 s to compute 20 paths in 100 x 100 grids with a 0.6% deviation to the optimal dissimilarity.