The minimum vulnerability problem on graphs

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.