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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.2022.12
URN: urn:nbn:de:0030-drops-174045
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17404/
Chattopadhyay, Arkadev ;
Ghosal, Utsab ;
Mukhopadhyay, Partha
Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product
Abstract
We establish an ε-sensitive hierarchy separation for monotone arithmetic computations. The notion of ε-sensitive monotone lower bounds was recently introduced by Hrubeš [Pavel Hrubeš, 2020]. We show the following:
- There exists a monotone polynomial over n variables in VNP that cannot be computed by 2^o(n) size monotone circuits in an ε-sensitive way as long as ε ≥ 2^(-Ω(n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone circuits but cannot be computed by any monotone arithmetic branching program (ABP) of n^o(log n) size, even in an ε-sensitive fashion as long as ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone ABPs but cannot be computed in n^o(log n) size by monotone formulas even in an ε-sensitive way, when ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by width-4 polynomial size monotone arithmetic branching programs (ABPs) but cannot be computed in 2^o(n^{1/d}) size by monotone, unbounded fan-in formulas of product depth d even in an ε-sensitive way, when ε ≥ 2^(-Ω(n^{1/d})). This yields an ε-sensitive separation of constant-depth monotone formulas and constant-width monotone ABPs. The novel feature of our separations is that in each case the polynomial exhibited is obtained from a graph inner-product polynomial by choosing an appropriate graph topology. The closely related graph inner-product Boolean function for expander graphs was invented by Hayes [Thomas P. Hayes, 2011], also independently by Pitassi [Toniann Pitassi, 2009], in the context of best-partition multiparty communication complexity.
BibTeX - Entry
@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2022.12,
author = {Chattopadhyay, Arkadev and Ghosal, Utsab and Mukhopadhyay, Partha},
title = {{Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17404},
URN = {urn:nbn:de:0030-drops-174045},
doi = {10.4230/LIPIcs.FSTTCS.2022.12},
annote = {Keywords: Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations}
}
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Keywords: |
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Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations |
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Collection: |
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42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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14.12.2022 |
