Abstract
We establish an εsensitive hierarchy separation for monotone arithmetic computations. The notion of εsensitive 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 εsensitive 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 εsensitive 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 εsensitive way, when ε ≥ n^(Ω(log n)).
 There exists a polynomial over n variables that can be computed by width4 polynomial size monotone arithmetic branching programs (ABPs) but cannot be computed in 2^o(n^{1/d}) size by monotone, unbounded fanin formulas of product depth d even in an εsensitive way, when ε ≥ 2^(Ω(n^{1/d})). This yields an εsensitive separation of constantdepth monotone formulas and constantwidth monotone ABPs. The novel feature of our separations is that in each case the polynomial exhibited is obtained from a graph innerproduct polynomial by choosing an appropriate graph topology. The closely related graph innerproduct Boolean function for expander graphs was invented by Hayes [Thomas P. Hayes, 2011], also independently by Pitassi [Toniann Pitassi, 2009], in the context of bestpartition 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 InnerProduct}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {12:112:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772617},
ISSN = {18688969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17404},
URN = {urn:nbn:de:0030drops174045},
doi = {10.4230/LIPIcs.FSTTCS.2022.12},
annote = {Keywords: Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations}
}
Keywords: 

Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations 
Collection: 

42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) 
Issue Date: 

2022 
Date of publication: 

14.12.2022 