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New Lower Bounds for epsilon-Nets

Authors Andrey Kupavskii, Nabil Mustafa, János Pach

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Andrey Kupavskii
Nabil Mustafa
János Pach

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Andrey Kupavskii, Nabil Mustafa, and János Pach. New Lower Bounds for epsilon-Nets. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.54

Abstract

Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-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 epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (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 R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.

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Keywords
  • epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces

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