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Journal of Combinatorial Optimization

, Volume 26, Issue 4, pp 709–722 | Cite as

Finding paths with minimum shared edges

  • Masoud T. Omran
  • Jörg-Rüdiger Sack
  • Hamid Zarrabi-Zadeh
Article

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.

Keywords

Minimum shared edges problem Approximation algorithm Inapproximability Heuristic algorithms 

Notes

Acknowledgements

The authors would like to thank Anil Maheshwari and Peter Widmayer for helpful discussions.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Masoud T. Omran
    • 1
  • Jörg-Rüdiger Sack
    • 1
  • Hamid Zarrabi-Zadeh
    • 2
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Computer EngineeringSharif University of TechnologyTehranIran

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