research-article

Approaching the Chasm at Depth Four

Authors:

Pritish Kamath,

Ramprasad SaptharishiAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 6

Article No.: 33, Pages 1 - 16

https://doi.org/10.1145/2629541

Published: 17 December 2014 Publication History

Abstract

Agrawal and Vinay [2008], Koiran [2012], and Tavenas [2013] have recently shown that an exp (ω(√n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to a superpolynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.

We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp,(Ω(√n)).

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Cited By

Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649760
Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Limaye NSrinivasan STavenas S(2022)Guest ColumnACM SIGACT News10.1145/3544979.354498953:2(40-62)Online publication date: 25-Jul-2022
https://dl.acm.org/doi/10.1145/3544979.3544989
Show More Cited By

Index Terms

Approaching the Chasm at Depth Four
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Representation of mathematical objects
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 61, Issue 6

November 2014

285 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2700084

Editor:
Victor Vianu
University of California, San Diego

Issue’s Table of Contents

Copyright © 2014 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 December 2014

Accepted: 01 January 2014

Revised: 01 January 2014

Received: 01 September 2013

Published in JACM Volume 61, Issue 6

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Cited By

Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649760
Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Limaye NSrinivasan STavenas S(2022)Guest ColumnACM SIGACT News10.1145/3544979.354498953:2(40-62)Online publication date: 25-Jul-2022
https://dl.acm.org/doi/10.1145/3544979.3544989
Tavenas SLimaye NSrinivasan SLeonardi SGupta A(2022)Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplicationProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520044(416-425)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520044
Limaye NSrinivasan STavenas S(2022)Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00083(804-814)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00083
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Sero DGarachon IHermens ELiere RBatenburg K(2021)The Study of Three-Dimensional Fingerprint Recognition in Cultural HeritageJournal on Computing and Cultural Heritage 10.1145/346134114:4(1-20)Online publication date: 16-Jul-2021
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