The minimum vulnerability problem on specific graph classes

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

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1288-1304
Number of pages17
JournalJournal of Combinatorial Optimization
Issue number4
Publication statusPublished - 2016 Nov 1


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

ASJC Scopus subject areas

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics


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