License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.38
URN: urn:nbn:de:0030-drops-71991
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7199/
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Dutta, Kunal ; Ghosh, Arijit ; Jartoux, Bruno ; Mustafa, Nabil H.

Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

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Abstract

The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity.
In this paper we present several new results and applications related to packings:

* an optimal lower bound for shallow packings,

* improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry,

* we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and

* simplifying and generalizing one of the main technical tools in [Fox et al. , J. of the EMS, to appear].

BibTeX - Entry

@InProceedings{dutta_et_al:LIPIcs:2017:7199,
  author =	{Kunal Dutta and Arijit Ghosh and Bruno Jartoux and Nabil H. Mustafa},
  title =	{{Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7199},
  URN =		{urn:nbn:de:0030-drops-71991},
  doi =		{10.4230/LIPIcs.SoCG.2017.38},
  annote =	{Keywords: Epsilon-nets, Haussler's packing lemma, Mnets, shallow-cell complexity, shallow packing lemma}
}

Keywords: Epsilon-nets, Haussler's packing lemma, Mnets, shallow-cell complexity, shallow packing lemma
Collection: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 20.06.2017

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