Abstract
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.
Major parts of this work done while all authors were with TU Berlin. A full version of this paper is available at http://arxiv.org/abs/1705.03673.
T. Fluschnik—Supported by the DFG, project DAMM (NI 369/13-2).
M. Sorge—Supported by the DFG, project DAPA (NI 369/12-2), the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11 and by the Israel Science Foundation (grant no. 551145/14).
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Fluschnik, T., Morik, M., Sorge, M. (2017). The Complexity of Routing with Few Collisions. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_21
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