Skip to main content
Log in

The minimum vulnerability problem on specific graph classes

Journal of Combinatorial Optimization Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Aoki Y, Halldórsson BV, Halldórsson MM, Ito T, Konrad C, Zhou X (2014) The minimum vulnerability problem on graphs. In: Proc. COCOA 2014, LNCS 8881, pp. 299–313

  • Assadi S, Emamjomeh-Zadeh E, Norouzi-Fard A, Yazdanbod S, Zarrabi-Zadeh H (2012) The minimum vulnerability problem. In: Proc. ISAAC 2012, LNCS 7676, pp. 382–391

  • Bodlaender HL (1996) A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J Comput 25:1305–1317

    Article  MathSciNet  MATH  Google Scholar 

  • Brandstädt A, Le VB, Spinrad JP (1999) Graph classes: a survey. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Even G, Kortsarz G, Slany W (2005) On network design problems: fixed cost flows and the covering Steiner problem. ACM Trans Algorithms 1:74–101

    Article  MathSciNet  MATH  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco

    MATH  Google Scholar 

  • Gavril F (1974) The intersection graphs of subtrees in trees are exactly the chordal graphs. J Comb Theory Ser B 16:47–56

    Article  MathSciNet  MATH  Google Scholar 

  • Goldberg AV, Rao S (1998) Beyond the flow decomposition barrier. J ACM 45:783–797

    Article  MathSciNet  MATH  Google Scholar 

  • Isobe S, Zhou X, Nishizeki T (1999) A polynomial-time algorithm for finding total colorings of partial \(k\)-trees. Int J Found Comput Sci 10:171–194

    Article  MathSciNet  MATH  Google Scholar 

  • Krumke SO, Noltemeier H, Schwarz S, Wirth H-C, Ravi R (1998) Flow improvement and network flows with fixed costs. Proc OR 1998:158–167

    MathSciNet  MATH  Google Scholar 

  • Nishizeki T, Vygen J, Zhou X (2001) The edge-disjoint path problem is NP-complete for series-parallel graphs. Discret Appl Math 115:177–186

    Article  MathSciNet  MATH  Google Scholar 

  • Omran MT, Sack J-R, Zarrabi-Zadeh H (2013) Finding paths with minimum shared edges. J Combin Optim 26:709–722

    Article  MathSciNet  MATH  Google Scholar 

  • Spinrad JP (2003) Efficient graph representations. American Mathematical Society, Providence

    Book  MATH  Google Scholar 

  • Yang B, Yang M, Wang J, Zheng SQ (2005a) Minimum cost paths subject to minimum vulnerability for reliable communications. Proc ISPAN 2005:334–339

  • Yang M, Wang J, Qi X, Jiang Y (2005b) On finding the best partial multicast protection tree under dual-homing architecture. In: Proc IEEE HPSR, pp. 128–132

  • Ye ZQ, Li YM, Lu HQ, Zhou X (2013) Finding paths with minimum shared edges in graphs with bounded treewidth. Proc FCS 2013:40–46

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takehiro Ito.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aoki, Y., Halldórsson, B.V., Halldórsson, M.M. et al. The minimum vulnerability problem on specific graph classes. J Comb Optim 32, 1288–1304 (2016). https://doi.org/10.1007/s10878-015-9950-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-015-9950-2

Keywords

Navigation