Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Kupavskii, Andrey; Mustafa, Nabil; Pach, JŠnos http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-59467
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New Lower Bounds for epsilon-Nets

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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.

BibTeX - Entry

@InProceedings{kupavskii_et_al:LIPIcs:2016:5946,
  author =	{Andrey Kupavskii and Nabil Mustafa and J{\'a}nos Pach},
  title =	{{New Lower Bounds for epsilon-Nets}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{54:1--54:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5946},
  URN =		{urn:nbn:de:0030-drops-59467},
  doi =		{10.4230/LIPIcs.SoCG.2016.54},
  annote =	{Keywords: epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces}
}

Keywords: epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces
Seminar: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue date: 2016
Date of publication: 2016


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