The Minimum Vulnerability Problem

Authors:
Journal: Algorithmica, 70(4), 2014.
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 k/2-approximation algorithm for this problem, improving the best previous approximation factor of k−1 available for the problem. 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 a 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 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.
Journal version:
BibTex: