# The minimum vulnerability problem on specific graph classes

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## 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.

## 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. 299–313Google Scholar
- 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. 382–391Google Scholar
- Bodlaender HL (1996) A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J Comput 25:1305–1317MathSciNetCrossRefMATHGoogle Scholar
- Brandstädt A, Le VB, Spinrad JP (1999) Graph classes: a survey. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
- Even G, Kortsarz G, Slany W (2005) On network design problems: fixed cost flows and the covering Steiner problem. ACM Trans Algorithms 1:74–101MathSciNetCrossRefMATHGoogle Scholar
- Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San FranciscoMATHGoogle Scholar
- Gavril F (1974) The intersection graphs of subtrees in trees are exactly the chordal graphs. J Comb Theory Ser B 16:47–56MathSciNetCrossRefMATHGoogle Scholar
- Goldberg AV, Rao S (1998) Beyond the flow decomposition barrier. J ACM 45:783–797MathSciNetCrossRefMATHGoogle 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 10:171–194MathSciNetCrossRefMATHGoogle Scholar
- Krumke SO, Noltemeier H, Schwarz S, Wirth H-C, Ravi R (1998) Flow improvement and network flows with fixed costs. Proc OR 1998:158–167MathSciNetMATHGoogle Scholar
- Nishizeki T, Vygen J, Zhou X (2001) The edge-disjoint path problem is NP-complete for series-parallel graphs. Discret Appl Math 115:177–186MathSciNetCrossRefMATHGoogle Scholar
- Omran MT, Sack J-R, Zarrabi-Zadeh H (2013) Finding paths with minimum shared edges. J Combin Optim 26:709–722MathSciNetCrossRefMATHGoogle Scholar
- Spinrad JP (2003) Efficient graph representations. American Mathematical Society, ProvidenceCrossRefMATHGoogle Scholar
- Yang B, Yang M, Wang J, Zheng SQ (2005a) Minimum cost paths subject to minimum vulnerability for reliable communications. Proc ISPAN 2005:334–339Google Scholar
- 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. 128–132Google Scholar
- Ye ZQ, Li YM, Lu HQ, Zhou X (2013) Finding paths with minimum shared edges in graphs with bounded treewidth. Proc FCS 2013:40–46Google Scholar