The Complexity of Routing with Few Collisions

  • Till FluschnikEmail author
  • Marco Morik
  • Manuel Sorge
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph G with two distinct terminal vertices and two positive integers p and k, the question is whether one can connect the terminals by at least p routes (e.g. paths) such that at most k edges are time-wise shared among them. We study three types of routes: traverse each vertex at most once (paths), each edge at most once (trails), or no such restrictions (walks). We prove that for paths and trails the problem is \({\text {NP}}\)-complete on undirected and directed graphs even if k is constant or the maximum vertex degree in the input graph is constant. For walks, however, it is solvable in polynomial time on undirected graphs for arbitrary k and on directed graphs if k is constant. We additionally study for all route types a variant of the problem where the maximum length of a route is restricted by some given upper bound. We prove that this length-restricted variant has the same complexity classification with respect to paths and trails, but for walks it becomes \({\text {NP}}\)-complete on undirected graphs.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Ben Gurion University of the NegevBeershebaIsrael

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