The minimum vulnerability problem on specific graph classes

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

研究成果: Article査読

3 被引用数 (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.

ジャーナルJournal of Combinatorial Optimization
出版ステータスPublished - 2016 11 1

ASJC Scopus subject areas

  • コンピュータ サイエンスの応用
  • 離散数学と組合せ数学
  • 制御と最適化
  • 計算理論と計算数学
  • 応用数学


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