The Minimum Vulnerability Problem

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.