Partially Disjoint Shortest Paths and Near-Shortest Paths Trees

One of the ways to increase communication reliability is by sending k duplicate messages along different routes. This gives rise to the problem of finding k shortest paths between a given source and destination. An unconstrained solution of the k shortest paths problem may output paths that overlap in almost all edges. Clearly, using such paths will have an adverse impact on the communication reliability. On the other extreme, a solution of k independent shortest paths, which are paths that share neither an edge nor an intermediate node may not be realistic for several reasons: such paths may not exist, if they exist they may be very long compared to the shortest path, and the computational effort of finding such paths may be prohibitive. This motivated us to investigate the intermediate case in which the number of edges that are not shared among any two paths in the output k paths is parameterized. We explore both exactly shortest paths and near-shortest paths. Our results are also generalized to the case of multi-criteria prioritized weights. Next, we consider the related albeit different problem of computing the k shortest paths trees, which are the k spanning trees with minimum total path length. This problem was introduced by Sedeño-Noda and González-Martín (2010). They solved it using a greedy algorithm and proved its correctness using linear programming theory. We provide an alternative, combinatorial and simpler proof of the correctness of the same greedy algorithm. We believe that the combinatorial approach can lead to a better understanding and possible extensions of the related results.