We consider the problems of minimum-cost design and augmentation of directed network clusters that have diameter 2 and maintain the same diameter after the deletion of up to R elements (nodes or arcs) anywhere in the cluster. The property of a network to maintain not only the overall connectivity, but also the same diameter after the deletion of multiple nodes/arcs is referred to as strong attack tolerance. This paper presents the proof of NP-completeness of the decision version of the problem, derives tight theoretical bounds, as well as develops a heuristic algorithm for the considered problems, which are extremely challenging to solve to optimality even for small networks. Computational experiments suggest that the proposed heuristic algorithm does identify high-quality near-optimal solutions; moreover, in the special case of undirected networks with identical arc construction costs, the algorithm provably produces an exact optimal solution to strongly attack-tolerant two-hop network design problem, regardless of the network size.
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The parameter β of the generated power-law graphs was chosen to approximately match the number of arcs in the corresponding uniform graphs with the same number of nodes. This procedure was performed for comparison reasons. It is described in more detail in Veremyev and Boginski (2012a).
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This work was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA).
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Pastukhov, G., Veremyev, A., Boginski, V. et al. Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks. J Comb Optim 27, 462–486 (2014). https://doi.org/10.1007/s10878-012-9523-6
- Network design
- Strong attack tolerance
- R-Robust 2-clubs
- Combinatorial optimization