eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-06-10
54:1
54:16
10.4230/LIPIcs.SoCG.2016.54
article
New Lower Bounds for epsilon-Nets
Kupavskii, Andrey
Mustafa, Nabil
Pach, János
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol051-socg2016/LIPIcs.SoCG.2016.54/LIPIcs.SoCG.2016.54.pdf
epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces