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Journal of Combinatorial Optimization

, Volume 32, Issue 4, pp 1288–1304 | Cite as

The minimum vulnerability problem on specific graph classes

  • Yusuke Aoki
  • Bjarni V. Halldórsson
  • Magnús M. Halldórsson
  • Takehiro Ito
  • Christian Konrad
  • Xiao Zhou
Article

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 any 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 for 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 is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete 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.

Keywords

Bounded treewidth graph Chordal graph Fixed parameter tractability Graph algorithm Minimum vulnerability problem 

Notes

Acknowledgments

Magnús M. Halldórsson and Christian Konrad are supported by Icelandic Research Fund Grant-of-Excellence No. 120032011. Takehiro Ito is partially supported by JSPS KAKENHI 25330003.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Yusuke Aoki
    • 1
  • Bjarni V. Halldórsson
    • 3
  • Magnús M. Halldórsson
    • 3
  • Takehiro Ito
    • 1
    • 2
  • Christian Konrad
    • 3
  • Xiao Zhou
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.CREST, JSTKawaguchiJapan
  3. 3.School of Computer ScienceReykjavík UniversityReykjavíkIceland

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