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Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product

Authors Arkadev Chattopadhyay, Utsab Ghosal, Partha Mukhopadhyay



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Author Details

Arkadev Chattopadhyay
  • TIFR, Mumbai, India
Utsab Ghosal
  • Chennai Mathematical Institute, India
Partha Mukhopadhyay
  • Chennai Mathematical Institute, India

Acknowledgements

We thank the anonymous reviewers for their feedback.

Cite AsGet BibTex

Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay. Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 12:1-12:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSTTCS.2022.12

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity
  • Discrepancy
  • Lower Bounds
  • Monotone Computations

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