research-article

Lower bounds for monotone arithmetic circuits via communication complexity

Authors:

Arkadev Chattopadhyay,

Partha MukhopadhyayAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 786 - 799

https://doi.org/10.1145/3406325.3451069

Published: 15 June 2021 Publication History

Abstract

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first improvement to this classical result: we construct a family of polynomials P_n in n variables, each of its monomials has non-negative coefficient, such that P_n can be computed by a polynomial-size depth-three formula but every monotone circuit computing it has size 2^{Ω(n^1/4/log(n))}.

The polynomial P_n embeds the SINK∘ XOR function devised recently by Chattopadhyay, Mande and Sherif (2020) to refute the Log-Approximate-Rank Conjecture in communication complexity. To prove our lower bound for P_n, we develop a general connection between corruption of combinatorial rectangles by any function f ∘ XOR and corruption of product polynomials by a certain polynomial P^f that is an arithmetic embedding of f. This connection should be of independent interest.

Using further ideas from communication complexity, we construct another family of set-multilinear polynomials f_n,m such that both F_n,m − є· f_n,m and F_n,m + є· f_n,m have monotone circuit complexity 2^Ω(n/log(n)) if є ≥ 2^{− Ω( m )} and F_n,m ∏_i=1ⁿ (x_i,1 +⋯+x_i,m), with m = O( n/logn ). The polynomials f_n,m have 0/1 coefficients and are in VNP. Proving such lower bounds for monotone circuits has been advocated recently by Hrubeš (2020) as a first step towards proving lower bounds against general circuits via his new approach.

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Cited By

Komarath BPandey ARahul C(2023)Monotone Arithmetic Complexity of Graph Homomorphism PolynomialsAlgorithmica10.1007/s00453-023-01108-085:9(2554-2579)Online publication date: 11-Mar-2023
https://dl.acm.org/doi/10.1007/s00453-023-01108-0
Aiswarya CArvind VSaurabh S(2022)Theory research in IndiaCommunications of the ACM10.1145/355049665:11(88-93)Online publication date: 20-Oct-2022
https://dl.acm.org/doi/10.1145/3550496

Index Terms

Lower bounds for monotone arithmetic circuits via communication complexity
1. Theory of computation
  1. Computational complexity and cryptography
    1. Algebraic complexity theory

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cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
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Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

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Published: 15 June 2021

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MATRICS grant of the Science and Engineering Research Board, DST, India
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STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 21 - 25, 2021

Virtual, Italy

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Cited By

Komarath BPandey ARahul C(2023)Monotone Arithmetic Complexity of Graph Homomorphism PolynomialsAlgorithmica10.1007/s00453-023-01108-085:9(2554-2579)Online publication date: 11-Mar-2023
https://dl.acm.org/doi/10.1007/s00453-023-01108-0
Aiswarya CArvind VSaurabh S(2022)Theory research in IndiaCommunications of the ACM10.1145/355049665:11(88-93)Online publication date: 20-Oct-2022
https://dl.acm.org/doi/10.1145/3550496

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