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On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

  • Waldo GálvezEmail author
  • Fabrizio Grandoni
  • Afrouz Jabal Ameli
  • Krzysztof Sornat
Conference paper
  • 83 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11926)

Abstract

In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum cardinality whose addition to the graph makes it \((k+1)\)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case \(k=1\) (a.k.a. the Tree Augmentation Problem or TAP) or \(k=2\) (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, nothing better is known for CacAP (hence for k-Connectivity Augmentation in general).

As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is \(\mathrm {APX}\)-hard.

In this paper we present a combinatorial \(\left( \frac{3}{2}+\varepsilon \right) \)-approximation for CycAP, for any constant \(\varepsilon >0\). We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

Keywords

Approximation algorithms Connectivity Augmentation Cactus Augmentation Cycle Augmentation 

Notes

Acknowledgments

We would like to thank the anonymous reviewers for their helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Waldo Gálvez
    • 1
    Email author
  • Fabrizio Grandoni
    • 1
  • Afrouz Jabal Ameli
    • 1
  • Krzysztof Sornat
    • 2
  1. 1.IDSIALuganoSwitzerland
  2. 2.University of WrocławWrocławPoland

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