The Minimum Vulnerability Problem on Graphs

Abstract

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

Magnús M. Halldórsson 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.