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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.29
URN: urn:nbn:de:0030-drops-66526
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6652/
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Ezra, Esther ; Lovett, Shachar

On the Beck-Fiala Conjecture for Random Set Systems

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Abstract

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Sigma), where each element x in X lies in t randomly selected sets of Sigma, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |\Sigma| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

BibTeX - Entry

@InProceedings{ezra_et_al:LIPIcs:2016:6652,
  author =	{Esther Ezra and Shachar Lovett},
  title =	{{On the Beck-Fiala Conjecture for Random Set Systems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{29:1--29:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6652},
  URN =		{urn:nbn:de:0030-drops-66526},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.29},
  annote =	{Keywords: Discrepancy theory, Beck-Fiala conjecture, Random set systems}
}

Keywords: Discrepancy theory, Beck-Fiala conjecture, Random set systems
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
Issue Date: 2016
Date of publication: 06.09.2016


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