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Flexible Graph Connectivity

Approximating network design problems between 1- and 2-connectivity

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Abstract

We introduce and study the problem Flexible Graph Connectivity, which in contrast to many classical connectivity problems features a non-uniform failure model. We distinguish between safe and unsafe resources and postulate that failures can only occur among the unsafe resources. Given an undirected edge-weighted graph and a set of unsafe edges, the task is to find a minimum-cost subgraph that remains connected after removing at most k unsafe edges. We give constant-factor approximation algorithms for this problem for \(k = 1\) as well as for unit costs and \(k \ge 1\). Our approximation guarantees are close to the known best bounds for special cases, such as the 2-edge-connected spanning subgraph problem and the tree augmentation problem. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min–max–min optimization problem.

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Notes

  1. To see that WTAP is a special case of FGC suppose that the unsafe edges form a spanning tree T of cost zero of the input graph. Then an optimal solution is a minimum-cost tree augmentation of T.

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Authors and Affiliations

  1. Department of Mathematics, ETH, Zurich, Switzerland

    David Adjiashvili

  2. Fakultät für Mathematik, TU Dortmund University, Dortmund, Germany

    Felix Hommelsheim

  3. Laboratoire G-SCOP, Grenoble INP, Univ. Grenoble-Alpes, Grenoble, France

    Moritz Mühlenthaler

Authors
  1. David Adjiashvili
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  2. Felix Hommelsheim
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  3. Moritz Mühlenthaler
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Correspondence to Moritz Mühlenthaler.

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A preliminary version of this article appeared in the proceedings of IPCO 2020 [4]

F. Hommelsheim: Research partially supported by the German Research Foundation (DFG), RTG 1855.

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Adjiashvili, D., Hommelsheim, F. & Mühlenthaler, M. Flexible Graph Connectivity. Math. Program. 192, 409–441 (2022). https://doi.org/10.1007/s10107-021-01664-9

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