License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2021.13
URN: urn:nbn:de:0030-drops-155245
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15524/
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Chattopadhyay, Arkadev ; Garg, Ankit ; Sherif, Suhail

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

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LIPIcs-FSTTCS-2021-13.pdf (0.7 MB)

Abstract

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

BibTeX - Entry

@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2021.13,
  author =	{Chattopadhyay, Arkadev and Garg, Ankit and Sherif, Suhail},
  title =	{{Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czy, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/15524},
  URN =		{urn:nbn:de:0030-drops-155245},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.13},
  annote =	{Keywords: Approximate Rank, Randomized Parity Decision Trees, Randomized Communication Complexity, XOR functions, Subspace Designs}
}

Keywords: Approximate Rank, Randomized Parity Decision Trees, Randomized Communication Complexity, XOR functions, Subspace Designs
Collection: 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)
Issue Date: 2021
Date of publication: 29.11.2021

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