Algorithmica

, Volume 70, Issue 4, pp 718–731 | Cite as

The Minimum Vulnerability Problem

  • Sepehr Assadi
  • Ehsan Emamjomeh-Zadeh
  • Ashkan Norouzi-Fard
  • Sadra Yazdanbod
  • Hamid Zarrabi-Zadeh
Article

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 \({\lfloor {k/2}\rfloor }\)-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 \(O(\log n)\) 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 a 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

Network design Shared edges Approximation algorithms  Inapproximability 

References

  1. 1.
    Assadi, S., Emamjomeh-Zadeh, E., Norouzi-Fard, A., Yazdanbod, S., Zarrabi-Zadeh, H.: The minimum vulnerability problem. In: Proceedings of the 23rd International Symposium on Algorithms and Computation, Volume 7676 of Lecture Notes in Computer Science, pp. 382–391 (2012)Google Scholar
  2. 2.
    Carr, R.D., Fleischer, L.K., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 106–115 (2000)Google Scholar
  3. 3.
    Chakrabarty, D., Chekuri, C., Khanna, S., Korula, N.: Approximability of capacitated network design. In: Proceedings of the 15th International Conference on Integer Programming and Combinatoral Optimization, Volume 6655 of Lecture Notes in Computer Science, pp. 78–91 (2011)Google Scholar
  4. 4.
    Chakrabarty, D., Krishnaswamy, R., Li, S., Narayanan, S.: Capacitated network design on undirected graphs. In: Proceedings of the 16th International Workshop on Approximation Algorithms, Volume 8096 of Lecture Notes in Computer Science, pp. 71–80 (2013)Google Scholar
  5. 5.
    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
  6. 6.
    Franklin, M.K.: Complexity and security of distributed protocols. Ph.D. thesis, Depatment of Computer Science, Columbia University (1994)Google Scholar
  7. 7.
    Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)MATHGoogle Scholar
  8. 8.
    Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms 37(1), 66–84 (2000)Google Scholar
  9. 9.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM 45(5), 783–797 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hajiaghayi, M., Khandekar, R., Kortsarz, G., Nutov, Z.: On the fixed cost \(k\)-flow problem and related problems. arXiv:1108.1176 (2011)
  12. 12.
    Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Struct. Algorithms 20(3), 465–482 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Krumke, S.O., Noltemeier, H., Schwarz, S., Wirth, H.-C., Ravi, R.: Flow improvement and network flows with fixed costs. In: Proceedings of the International Conference on Operation Research: OR-98, pp. 158–167 (1998)Google Scholar
  14. 14.
    Omran, M.T., Sack, J.-R., Zarrabi-Zadeh, H.: Finding paths with minimum shared edges. J. Comb. Optim. 26(4), 709–722 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    Yang, B., Yang, M., Wang, J., Zheng, S.Q.: Minimum cost paths subject to minimum vulnerability for reliable communications. In: Proceedings of the 8th International Symposium on Parallel Architectures, Algorithms and Networks, pp. 334–339. IEEE Computer Society (2005)Google Scholar
  17. 17.
    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. Netw. 18(5), 1436–1449 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sepehr Assadi
    • 1
  • Ehsan Emamjomeh-Zadeh
    • 1
  • Ashkan Norouzi-Fard
    • 1
  • Sadra Yazdanbod
    • 1
  • Hamid Zarrabi-Zadeh
    • 1
  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran

Personalised recommendations