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.APPROX-RANDOM.2016.24
URN: urn:nbn:de:0030-drops-66475
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Boppana, Ravi ; Håstad, Johan ; Lee, Chin Ho ; Viola, Emanuele

Bounded Independence vs. Moduli

LIPIcs-APPROX-RANDOM-2016-24.pdf (0.5 MB)


Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.

BibTeX - Entry

  author =	{Ravi Boppana and Johan Håstad and Chin Ho Lee and Emanuele Viola},
  title =	{{Bounded Independence vs. Moduli}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{24:1--24:9},
  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 =		{},
  URN =		{urn:nbn:de:0030-drops-66475},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.24},
  annote =	{Keywords: Bounded independence, Modulus}

Keywords: Bounded independence, Modulus
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
Issue Date: 2016
Date of publication: 06.09.2016

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