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The Minimum Vulnerability Problem on Graphs

  • Yusuke Aoki
  • Bjarni V. Halldórsson
  • Magnús M. Halldórsson
  • Takehiro Ito
  • Christian Konrad
  • Xiao Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

Abstract

Suppose that each edge \(e\) of an undirected graph \(G\) is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of \(e\), respectively. Then, we consider the problem of finding \(k\) paths in \(G\) between two prescribed vertices with the minimum total cost; each edge \(e\) can be shared without cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost of \(e\). This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem remains NP-hard even for bipartite series-parallel graphs and for threshold graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.

References

  1. 1.
    Assadi, S., Emamjomeh-Zadeh, E., Norouzi-Fard, A., Yazdanbod, S., Zarrabi-Zadeh, H.: The minimum vulnerability problem. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 382–391. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Boutiche, M.A., Ait Haddadène, H., Le Thi, H.A.: Maintaining graph properties of weakly chordal graphs. Appl. Math. Sci. 6, 765–778 (2012)MATHMathSciNetGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)CrossRefMATHGoogle 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, 74–101 (2005)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  7. 7.
    Isobe, S., Zhou, X., Nishizeki, T.: A polynomial-time algorithm for finding total colorings of partial \(k\)-trees. Int. J. Found. Comput. Sci. 10, 171–194 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Krumke, S.O., Noltemeier, H., Schwarz, S., Wirth, H.-C., Ravi, R.: Flow improvement and network flows with fixed costs. In: Proceedings of OR, vol. 1998, pp. 158–167 (1998)Google Scholar
  9. 9.
    Omran, M.T., Sack, J.-R., Zarrabi-Zadeh, H.: Finding paths with minimum shared edges. J. Comb. Optim. 26, 709–722 (2011)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ye, Z.Q., Li, Y.M., Lu, H.Q., Zhou, X.: Finding paths with minimum shared edges in graphs with bounded treewidth. In: Proceedings of FCS vol. 2013, pp. 40–46 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yusuke Aoki
    • 1
  • Bjarni V. Halldórsson
    • 2
  • Magnús M. Halldórsson
    • 2
  • Takehiro Ito
    • 1
  • Christian Konrad
    • 2
  • Xiao Zhou
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.School of Computer ScienceReykjavík UniversityReykjavikIceland

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