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Near-Optimal Lower Bounds for ε-Nets for Half-Spaces and Low Complexity Set Systems

  • Andrey Kupavskii
  • Nabil H. Mustafa
  • János Pach
Chapter

Abstract

Following groundbreaking work by Haussler and Welzl (1987), the use of small ε-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest ε-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Könemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in \(\mathbb{R}^{4}\) by a family of half-spaces such that the size of any ε-net for them is \(\Omega (\frac{1} {\epsilon } \log \frac{1} {\epsilon } )\) (Pach and Tardos).

The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in \(\mathbb{R}^{d},\) for any d ≥ 4, to show that the general upper bound, \(O(\frac{d} {\epsilon } \log \frac{1} {\epsilon } )\), of Haussler and Welzl for the size of the smallest ε-nets is tight.

Notes

Acknowledgements

We thank the anonymous referees for carefully reading our manuscript and for pointing out several problems in the presentation, including a mistake in the proof of Lemma 10. A preliminary version of this paper was accepted in SoCG 2016 (Symposium on Computational Geometry). We also thank the anonymous reviewers of the conference version for their feedback and valuable suggestions.

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Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  • Andrey Kupavskii
    • 1
    • 2
  • Nabil H. Mustafa
    • 3
  • János Pach
    • 2
    • 4
  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.EPFLLausanneSwitzerland
  3. 3.LIGM, Equipe A3SI, ESIEE ParisUniversité Paris-EstChamps-sur-MarneFrance
  4. 4.Rényi InstituteBudapestHungary

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