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Fast Exact Algorithms for Survivable Network Design with Uniform Requirements

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Abstract

We design exact algorithms for the following two problems in survivable network design: (i) designing a minimum cost network with a desired value of edge connectivity, which is called Minimum Weight \(\lambda \) -connected Spanning Subgraph and (ii) augmenting a given network to a desired value of edge connectivity at a minimum cost which is called Minimum Weight \(\lambda \) -connectivity Augmentation. It is easy to see that a minimum solution to these problems contains at most \(2 \lambda (n-1)\) edges. Using this fact one can design a brute-force algorithm which runs in time \(2^{{\mathcal {O}}(\lambda n \log n)}\), however no better algorithms were known previously. In this paper, we give the first single exponential time algorithm for these problems, i.e. running in time \(2^{{\mathcal {O}}(\lambda n)}\), for both undirected and directed networks. Our results are obtained via well known characterizations of \(\lambda \)-connected graphs, their connections to linear matroids and the recently developed technique of dynamic programming with representative sets.

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Notes

  1. We slightly abuse notation for the sake of clarity, as strictly speaking X and \(\textsf {Out}_{D^r_G}(v)\) are disjoint, since they are subsets of two different copies of the arc set.

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Correspondence to Akanksha Agrawal.

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A preliminary version of this article appeared at WADS 2017.

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Agrawal, A., Misra, P., Panolan, F. et al. Fast Exact Algorithms for Survivable Network Design with Uniform Requirements. Algorithmica 84, 2622–2641 (2022). https://doi.org/10.1007/s00453-022-00959-3

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