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Discrete & Computational Geometry

, Volume 57, Issue 4, pp 985–1011 | Cite as

Packing and Covering with Balls on Busemann Surfaces

  • Victor Chepoi
  • Bertrand Estellon
  • Guyslain NavesEmail author
Article

Abstract

In this note we prove that for any compact subset S of a Busemann surface \(({{\mathcal {S}}},d)\) (in particular, for any simple polygon with geodesic metric) and any positive number \(\delta \), the minimum number of closed balls of radius \(\delta \) with centers at \({\mathcal {S}}\) and covering the set S is at most 19 times the maximum number of disjoint closed balls of radius \(\delta \) centered at points of S: \(\nu (S)\le \rho (S)\le 19{}\nu (S)\), where \(\rho (S)\) and \(\nu (S)\) are the covering and the packing numbers of S by \({\delta }\)-balls. Busemann surfaces represent a far-reaching generalization not only of simple polygons, but also of Euclidean and hyperbolic planes and of all planar polygonal complexes of global non-positive curvature. Roughly speaking, a Busemann surface is a geodesic metric space homeomorphic to \({{\mathbb {R}}}^2\) in which the distance function is convex.

Keywords

Packing number Covering numbers Balls Busemann surfaces 

Notes

Acknowledgements

The authors would like to thank the referees of this paper for careful reading of the previous versions and many useful remarks.

References

  1. 1.
    Agarwal, P.K., Mustafa, N.H.: Independent set of intersection graphs of convex objects in 2D. Comput. Geom. 34(2), 83–95 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N.: Piercing \(d\)-intervals. Discrete Comput. Geom. 19(3), 333–334 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N.: Covering a hypergraph of subgraphs. Discrete Math. 257(2–3), 249–254 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bárány, I., Edmonds, J., Wolsey, L.A.: Packing and covering a tree by subtrees. Combinatorica 6(3), 221–233 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berge, C.: Hypergraphs. North-Holland Mathematical Library, vol. 45. North-Holland, Amsterdam (1989)Google Scholar
  6. 6.
    Böröczky, K.J.: Finite Packing and Covering. Cambridge Tracts in Mathematics, vol. 154. Cambridge University Press, Cambridge (2004)Google Scholar
  7. 7.
    Borradaile, G., Chambers, E.W.: Covering nearly surface-embedded graphs with a fixed number of balls. Discrete Comput. Geom. 51(4), 979–996 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bousquet, N.: Hitting Sets: VC-dimension and Multicut. PhD thesis, Université de Montpellier II (2013). https://tel.archives-ouvertes.fr/tel-01012106
  9. 9.
    Bousquet, N., Thomassé, S.: VC-dimension and Erdős–Pósa property. Discrete Math. 338(12), 2302–2317 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)Google Scholar
  11. 11.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cameron, P.J.: Problems from CGCS Luminy, May 2007. Eur. J. Combin. 31(2), 644–648 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chalopin, J., Chepoi, V., Naves, G.: Isometric embedding of Busemann surfaces into \(L_1\). Discrete Comput. Geom. 53(1), 16–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom. 48(2), 373–392 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chepoi, V., Estellon, B., Vaxès, Y.: On covering planar graphs with a fixed number of balls. Discrete Comput. Geom. 37, 237–244 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chepoi, V., Estellon, B.: Packing and covering \(\delta \)-hyperbolic spaces by balls. In: Charikar, M., et al. (eds.) Approximation, Randomization, and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 4627, pp. 59–73. Springer, Berlin (2007)CrossRefGoogle Scholar
  17. 17.
    Chepoi, V., Felsner, S.: Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Comput. Geom. 46(9), 1036–1041 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Clarkson, K.L.: Nearest neighbor queries in metric spaces. Discrete Comput. Geom. 22(1), 63–93 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cornuéjols, G.: Combinatorial Optimization: Packing and Covering. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74. SIAM, Philadelphia (2001)Google Scholar
  20. 20.
    Correa, J., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity. Discrete Comput. Geom. 53(2), 344–365 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, Cambridge (1999)Google Scholar
  22. 22.
    Gyárfás, A., Lehel, J.: Covering and coloring problems for relatives of intervals. Discrete Math. 55(2), 167–180 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hadwiger, H., Debrunner, H.: Kombinatorische Geometrie in der Ebene. Monographies de “L’Enseignement Mathématique”, No. 2. Institut de Mathématiques, Université Genève, Genève (1959)Google Scholar
  24. 24.
    Ivanov, S.: On Helly’s theorem in geodesic spaces. Electron. Res. Announc. Math. Sci. 21, 109–112 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Karasev, R.N.: Transversals for families of translates of a two-dimensional convex compact set. Discrete Comput. Geom. 24(2–3), 345–353 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kolmogorov, A.N., Tihomirov, V.M.: \(\varepsilon \)-Entropy and \(\varepsilon \)-Capacity of sets in function spaces. Am. Math. Soc. Transl. 17, 282–369 (1961)Google Scholar
  27. 27.
    Kulkarni, S.R.: On Metric entropy, Vapnik–Chervonenkis dimension, and learnability for a class of distributions. Technical Report 868 (1989). http://www.dtic.mil/dtic/tr/fulltext/2/a217331
  28. 28.
    Lorentz, G.G.: Metric entropy and approximation. Bull. Am. Math. Soc. 72(6), 903–937 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)Google Scholar
  31. 31.
    Papadopoulos, A.: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics, vol. 6. European Mathematical Society, Zürich (2005)Google Scholar
  32. 32.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(6), 611–626 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Schrijver, A.: Combinatorial Optimization, Vol. B. Algorithms and Combinatorics, vol. 24.B. Springer, Berlin (2003)Google Scholar
  34. 34.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)zbMATHGoogle Scholar
  35. 35.
    Vigan, I.: Packing and covering a polygon with geodesic disks. http://arxiv.org/abs/1311.6033 (2013)
  36. 36.
    Wegner, G.: Über eine kombinatorisch-geometrische Frage von Hadwiger and Debrunner. Isr. J. Math. 3(4), 187–198 (1965)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Victor Chepoi
    • 1
  • Bertrand Estellon
    • 1
  • Guyslain Naves
    • 1
    Email author
  1. 1.Aix Marseille Université, CNRS, LIF UMR 7279MarseilleFrance

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