Kupavskii, Andrey ;
Mustafa, Nabil ;
Pach, János
New Lower Bounds for epsilonNets
Abstract
Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilonnets 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 epsilonnets for set systems, as a function of their socalled shallowcell 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 halfspaces such that the size of any epsilonnet 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 halfspaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilonnets 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 epsilonNets}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {54:154:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5946},
URN = {urn:nbn:de:0030drops59467},
doi = {10.4230/LIPIcs.SoCG.2016.54},
annote = {Keywords: epsilonnets; lower bounds; geometric set systems; shallowcell complexity; halfspaces}
}
2016
Keywords: 

epsilonnets; lower bounds; geometric set systems; shallowcell complexity; halfspaces 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016)

Issue date: 

2016 
Date of publication: 

2016 