The Minimum Vulnerability Problem

  • Sepehr Assadi
  • Ehsan Emamjomeh-Zadeh
  • Ashkan Norouzi-Fard
  • Sadra Yazdanbod
  • Hamid Zarrabi-Zadeh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)

Abstract

We revisit the problem of finding k paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the k paths. We provide a [k/2]-approximation algorithm for this problem, improving the best previous approximation factor of k − 1. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of O(n3/4), where n is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be improved to \(O(\sqrt{n})\). While the problem is NP-hard, and even hard to approximate to within an O(logn) factor, we show that the problem is polynomially solvable when k is a constant. This settles an open problem posed by Omran et al.  regarding the complexity of the problem for small values of k. We present most of our results in a more general form where each edge of the graph has a sharing cost and a sharing capacity, and there is vulnerability parameter r that determines the number of times an edge can be used among different paths before it is counted as a shared/vulnerable edge.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Carr, R.D., Fleischer, L.K., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pp. 106–115 (2000)Google Scholar
  2. 2.
    Even, G., Kortsarz, G., Slany, W.: On network design problems: fixed cost flows and the covering steiner problem. ACM Trans. Algorithms 1(1), 74–101 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Franklin, M.K.: Complexity and security of distributed protocols. PhD thesis, Dept. of Computer Science, Columbia University (1994)Google Scholar
  4. 4.
    Garey, M., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman (1979)Google Scholar
  5. 5.
    Garg, N., Ravi, R., Konjevod, G.: A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms 37(1) (2000)Google Scholar
  6. 6.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM 45(5), 783–797 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Structures & Algorithms 20(3), 465–482 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Krumke, S.O., Noltemeier, H., Schwarz, S., Wirth, H.-C., Ravi, R.: Flow improvement and network flows with fixed costs. In: Proc. Internat. Conf. Oper. Res.: OR 1998, pp. 158–167 (1998)Google Scholar
  10. 10.
    Omran, M.T., Sack, J.-R., Zarrabi-Zadeh, H.: Finding Paths with Minimum Shared Edges. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 567–578. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Wang, J., Yang, M., Yang, B., Zheng, S.Q.: Dual-homing based scalable partial multicast protection. IEEE Trans. Comput. 55(9), 1130–1141 (2006)CrossRefGoogle Scholar
  12. 12.
    Williamson, D.P., Shmoys, D.B.: The design of approximation algorithms. Cambridge University Press (2011)Google Scholar
  13. 13.
    Yang, B., Yang, M., Wang, J., Zheng, S.Q.: Minimum cost paths subject to minimum vulnerability for reliable communications. In: Proc. 8th Internat. Symp. Parallel Architectures, Algorithms and Networks, ISPAN 2005, pp. 334–339. IEEE Computer Society (2005)Google Scholar
  14. 14.
    Zheng, S.Q., Wang, J., Yang, B., Yang, M.: Minimum-cost multiple paths subject to minimum link and node sharing in a network. IEEE/ACM Trans. Networking 18(5), 1436–1449 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sepehr Assadi
    • 1
  • Ehsan Emamjomeh-Zadeh
    • 1
  • Ashkan Norouzi-Fard
    • 1
  • Sadra Yazdanbod
    • 1
  • Hamid Zarrabi-Zadeh
    • 1
    • 2
  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran
  2. 2.Institute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations