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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(n 3/4), 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.

Keywords

Approximation Algorithm Approximation Factor Network Design Problem Steiner Tree Problem Sparse Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Structures & Algorithms 20(3), 465–482 (2002)MathSciNetzbMATHCrossRefGoogle 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

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