Finding Critical Regions and Region-Disjoint Paths in a Network

Due to their importance to society, communication networks should be built and operated to withstand failures. However, cost considerations make network providers less inclined to take robustness measures against failures that are unlikely to manifest, like several failures coinciding simultaneously in different geographic regions of their network. Considering networks embedded in a two-dimensional plane, we study the problem of finding a critical region-a part of the network that can be enclosed by a given elementary figure of predetermined size-whose destruction would lead to the highest network disruption. We determine that only a polynomial, in the input, number of nontrivial positions for such a figure needs to be considered and propose a corresponding polynomial-time algorithm. In addition, we consider region-aware network augmentation to decrease the impact of a regional failure. We subsequently address the region-disjoint paths problem, which asks for two paths with minimum total weight between a source (s) and a destination (d) that cannot both be cut by a single regional failure of diameter D (unless that failure includes s or d). We prove that deciding whether region-disjoint paths exist is NP-hard and propose a heuristic region-disjoint paths algorithm.