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DOI: 10.4230/LIPIcs.ICALP.2019.58
URN: urn:nbn:de:0030-drops-106340
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10634/
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Filmus, Yuval ; Hambardzumyan, Lianna ; Hatami, Hamed ; Hatami, Pooya ; Zuckerman, David

Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

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LIPIcs-ICALP-2019-58.pdf (0.5 MB)


Abstract

The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product measures on {0,1}^n, by combining their argument with a completely different argument that handles very biased input bits. We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube [0,1]^n (or, equivalently, on {1,...,n}^n) can be biased using coalitions of o(n) players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004. Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is o(log^* n), a coalition of o(n) players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on {0,1}^n. The argument of Russell et al. relies on the fact that a coalition of o(n) players can boost the expectation of any Boolean function from epsilon to 1-epsilon with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to mu_{1-1/n} shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

BibTeX - Entry

@InProceedings{filmus_et_al:LIPIcs:2019:10634,
  author =	{Yuval Filmus and Lianna Hambardzumyan and Hamed Hatami and Pooya Hatami and David Zuckerman},
  title =	{{Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{58:1--58:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10634},
  URN =		{urn:nbn:de:0030-drops-106340},
  doi =		{10.4230/LIPIcs.ICALP.2019.58},
  annote =	{Keywords: Boolean function analysis, coin flipping}
}

Keywords: Boolean function analysis, coin flipping
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 08.07.2019


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