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/
Chattopadhyay, Arkadev ;
Garg, Ankit ;
Sherif, Suhail
Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture
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: |
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Approximate Rank, Randomized Parity Decision Trees, Randomized Communication Complexity, XOR functions, Subspace Designs |
Collection: |
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41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) |
Issue Date: |
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2021 |
Date of publication: |
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29.11.2021 |