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Finding paths with minimum shared edges

Author

Listed:
  • Masoud T. Omran

    (Carleton University)

  • Jörg-Rüdiger Sack

    (Carleton University)

  • Hamid Zarrabi-Zadeh

    (Sharif University of Technology)

Abstract

Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of $2^{\log^{1-\varepsilon}n}$ , for any constant ε>0. On the positive side, we show that there exists a (k−1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design some heuristics to improve the quality of the output, and provide empirical results.

Suggested Citation

  • Masoud T. Omran & Jörg-Rüdiger Sack & Hamid Zarrabi-Zadeh, 2013. "Finding paths with minimum shared edges," Journal of Combinatorial Optimization, Springer, vol. 26(4), pages 709-722, November.
    • Handle: RePEc:spr:jcomop:v:26:y:2013:i:4:d:10.1007_s10878-012-9462-2
    DOI: 10.1007/s10878-012-9462-2
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    Cited by:

    1. Yusuke Aoki & Bjarni V. Halldórsson & Magnús M. Halldórsson & Takehiro Ito & Christian Konrad & Xiao Zhou, 2016. "The minimum vulnerability problem on specific graph classes," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1288-1304, November.

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