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Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

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Approximation and Online Algorithms (WAOA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12982))

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Abstract

Given a k-vertex-connected graph G and a set S of extra edges (links), the goal of the k-vertex-connectivity augmentation problem is to find a subset \(S'\) of S of minimum size such that adding \(S'\) to G makes it \((k+1)\)-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse.

In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for 2-vertex-connectivity augmentation, for the case in which G is a cycle. This is the first step for attacking the more general problem of augmenting a 2-connected graph.

Our algorithm is based on local search and attains an approximation ratio of 1.8703. To derive it, we prove novel results on the structure of minimal solutions.

Waldo Gálvez is supported by the European Research Council, Grant Agreement No. 691672, project APEG. Francisco Sanhueza-Matamala is partially supported by grants ANID-PFCHA/Magíster Nacional/2020- 22201780 and FONDECYT Regular 1190043. Francisco Sanhueza-Matamala and José A. Soto are partially supported by ANID via FONDECYT Regular 1181180 and PIA AFB170001.

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Notes

  1. 1.

    For \(k\in \mathbb {N}\), a k-connected graph is a graph \(G=(V,E)\) satisfying that, for any \(V'\subseteq V\) with \(|V|\le k-1\), G remains connected after the deletion of \(V'\). If the definition holds when replacing nodes by edges, the graph is said to be k-edge-connected.

  2. 2.

    A cactus is a 2-edge-connected graph where every edge belongs exactly to one cycle of the graph.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Technical University of Munich, Munich, Germany

    Waldo Gálvez

  2. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Francisco Sanhueza-Matamala & José A. Soto

  3. Centro de Modelamiento Matemático, IRL 2807 CNRS, Universidad de Chile, Santiago, Chile

    José A. Soto

Authors
  1. Waldo Gálvez
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  2. Francisco Sanhueza-Matamala
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  3. José A. Soto
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Corresponding author

Correspondence to Francisco Sanhueza-Matamala .

Editor information

Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Jochen Koenemann

  2. RWTH Aachen University, Aachen, Germany

    Britta Peis

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Gálvez, W., Sanhueza-Matamala, F., Soto, J.A. (2021). Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle. In: Koenemann, J., Peis, B. (eds) Approximation and Online Algorithms. WAOA 2021. Lecture Notes in Computer Science(), vol 12982. Springer, Cham. https://doi.org/10.1007/978-3-030-92702-8_1

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  • DOI: https://doi.org/10.1007/978-3-030-92702-8_1

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  • Online ISBN: 978-3-030-92702-8

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