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Packing and Covering δ-Hyperbolic Spaces by Balls

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Book cover Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

Abstract

We consider the problem of covering and packing subsets of δ-hyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δ-hyperbolic graph G and a positive number R, let γ(S,R) be the minimum number of balls of radius R covering S. It is known that computing γ(S,R) or approximating this number within a constant factor is hard even for 2-hyperbolic graphs. In this paper, using a primal-dual approach, we show how to construct in polynomial time a covering of S with at most γ(S,R) balls of (slightly larger) radius R + δ. This result is established in the general framework of δ-hyperbolic geodesic metric spaces and is extended to some other set families derived from balls. The covering algorithm is used to design better approximation algorithms for the augmentation problem with diameter constraints and for the k-center problem in δ-hyperbolic graphs.

This research was partly supported by the ANR grant BLAN06-1-138894 (projet OPTICOMB).

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, Universitée de la, Méditerranée, F-13288 Marseille Cedex 9, France

    Victor Chepoi & Bertrand Estellon

Authors
  1. Victor Chepoi
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  2. Bertrand Estellon
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Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

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Chepoi, V., Estellon, B. (2007). Packing and Covering δ-Hyperbolic Spaces by Balls. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_5

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  • DOI: https://doi.org/10.1007/978-3-540-74208-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

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