The Minimum Vulnerability Problem

Abstract

We revisit the problem of finding $k$ paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the $k$ paths. We provide a ${\lfloor {k/2}\rfloor }$-approximation algorithm for this problem, improving the best previous approximation factor of $k-1$. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of $O(n^{3/4})$, where $n$ is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be improved to $O(\sqrt{n})$. While the problem is NP-hard, and even hard to approximate to within an $O(\log n)$ factor, we show that the problem is polynomially solvable when $k$ is a constant. This settles an open problem posed by Omran et al. regarding the complexity of the problem for small values of $k$. We present most of our results in a more general form where each edge of the graph has a sharing cost and a sharing capacity, and there is a vulnerability parameter $r$ that determines the number of times an edge can be used among different paths before it is counted as a shared/vulnerable edge.