Packing and Covering δ-Hyperbolic Spaces by Balls

  • Authors:

  • Victor Chepoi;Bertrand Estellon

  • Affiliations:

  • Laboratoire d'Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, Universitée de la, Méditerranée, F-13288 Marseille Cedex 9, France;Laboratoire d'Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, Universitée de la, Méditerranée, F-13288 Marseille Cedex 9, France

  • Venue:
  • APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
  • Year:
  • 2007

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Abstract

We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs.