TY - JOUR

T1 - The minimum vulnerability problem on specific graph classes

AU - Aoki, Yusuke

AU - Halldórsson, Bjarni V.

AU - Halldórsson, Magnús M.

AU - Ito, Takehiro

AU - Konrad, Christian

AU - Zhou, Xiao

N1 - Funding Information:
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.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - Suppose that each edge e of an undirected graph G is associated with three nonnegative integers cost( e) , vul( e) and 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 vul( e) paths, and can be shared by more than vul( e) paths if we pay cost( e) , but cannot be shared by more than 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.

AB - Suppose that each edge e of an undirected graph G is associated with three nonnegative integers cost( e) , vul( e) and 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 vul( e) paths, and can be shared by more than vul( e) paths if we pay cost( e) , but cannot be shared by more than 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.

KW - Bounded treewidth graph

KW - Chordal graph

KW - Fixed parameter tractability

KW - Graph algorithm

KW - Minimum vulnerability problem

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U2 - 10.1007/s10878-015-9950-2

DO - 10.1007/s10878-015-9950-2

M3 - Article

AN - SCOPUS:84941341654

VL - 32

SP - 1288

EP - 1304

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 4

ER -