TY - JOUR

T1 - The minimum vulnerability problem on graphs

AU - Aoki, Yusuke

AU - Halldórsson, Bjarni V.

AU - Halldórsson, Magnús M.

AU - Ito, Takehiro

AU - Konrad, Christian

AU - Zhou, Xiao

N1 - Funding Information:
Magn?s M. Halld?rsson and Christian Konrad are supported by Icelandic Research Fund grant-of-excellence no. 120032011. Takehiro Ito: This work is partially supported by JSPS KAKENHI 25106504 and 25330003.
Publisher Copyright:
© Springer International Publishing Switzerland 2014.

PY - 2014/1/1

Y1 - 2014/1/1

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 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 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.

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 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 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.

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U2 - 10.1007/978-3-319-12691-3_23

DO - 10.1007/978-3-319-12691-3_23

M3 - Article

AN - SCOPUS:84996484196

VL - 8881

SP - 299

EP - 313

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -