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Approximating k-Edge-Connected Spanning Subgraphs via a Near-Linear Time LP Solver

Authors Parinya Chalermsook, Chien-Chung Huang, Danupon Nanongkai, Thatchaphol Saranurak, Pattara Sukprasert, Sorrachai Yingchareonthawornchai

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Author Details

Parinya Chalermsook
  • Aalto University, Espoo, Finland
Chien-Chung Huang
  • École Normale Supérieure, Paris, France
Danupon Nanongkai
  • University of Copenhagen, Denmark
  • KTH Royal Institute of Technology, Stockholm, Sweden
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Pattara Sukprasert
  • Northwestern University, Evanston, IL, USA
Sorrachai Yingchareonthawornchai
  • Aalto University, Espoo, Finland

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 759557 & 715672. Nanongkai and Saranurak were also partially supported by the Swedish Research Council (Reg. No. 2015-04659.). Chalermsook was also supported by the Academy Research Fellowship (grant number 310415).

Acknowledgements

We thank the 2021 Hausdorff Research Institute for Mathematics Program Discrete Optimization during which part of this work was developed. Parinya Chalermsook thanks Chandra Chekuri for clarifications of his FOCS 2017 paper and for giving some pointers. We thank Corinna Coupette for bringing [Michel X. Goemans et al., 1994] to our attention.

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Parinya Chalermsook, Chien-Chung Huang, Danupon Nanongkai, Thatchaphol Saranurak, Pattara Sukprasert, and Sorrachai Yingchareonthawornchai. Approximating k-Edge-Connected Spanning Subgraphs via a Near-Linear Time LP Solver. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.37

Abstract

In the k-edge-connected spanning subgraph (kECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to k link failures: Given an n-node m-edge graph with a cost function on the edges, our goal is to compute a minimum-cost k-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of 2ECSS. The fastest 2-approximation algorithm is however rather slow, taking O(mn k) time [Khuller, Vishkin, STOC'92]. A faster time complexity of O(n²) can be obtained, but with a higher approximation guarantee of (2k-1) [Gabow, Goemans, Williamson, IPCO'93].
Our main contribution is an algorithm that (1+ε)-approximates the optimal fractional solution in Õ(m/ε²) time (independent of k), which can be turned into a (2+ε) approximation algorithm that runs in time Õ(m/(ε²) + {k²n^{1.5}}/ε²) for (integral) kECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.

Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
Keywords
  • Approximation Algorithms
  • Data Structures

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