Critical regions and region-disjoint paths in a network

Due to the importance of communication networks to society, it is pertinent that these networks can withstand failures. Improving the robustness of a network usually requires installing redundant resources, which is very costly. Network providers are consequently 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. Protecting against single regional failures is more realistic. Network robustness, in terms of connectivity properties, also requires survivability algorithms to quickly reroute traffic affected by a network failure. In this paper, we consider a network embedded in a plane and study the problem of finding a circular region with radius r in that plane that would cause the biggest network degradation if all nodes within that particular region were to be destroyed. We propose a polynomial time algorithm for finding such critical regions. In addition, we develop a region-aware network augmentation technique to decrease the impact of a critical region failure. We subsequently consider 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 circular regional failure of radius r (unless that failure includes s and d). We prove that the region-disjoint paths problem is NP-hard and propose and evaluate a heuristic algorithm for it.