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Connected Vertex Covers in Dense Graphs

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Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (APPROX 2008, RANDOM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5171))

Abstract

We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worst-case ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of near-tight examples.

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Authors and Affiliations

  1. Université Libre de Bruxelles, CP212, B-1050, Brussels, Belgium

    Jean Cardinal & Eythan Levy

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  1. Jean Cardinal
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  2. Eythan Levy
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Ashish Goel Klaus Jansen José D. P. Rolim Ronitt Rubinfeld

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© 2008 Springer-Verlag Berlin Heidelberg

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Cardinal, J., Levy, E. (2008). Connected Vertex Covers in Dense Graphs. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_4

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  • DOI: https://doi.org/10.1007/978-3-540-85363-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85362-6

  • Online ISBN: 978-3-540-85363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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