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Monotone Circuit Lower Bounds from Robust Sunflowers.

Bruno Pasqualotto Cavalar1, Mrinal Kumar2, Benjamin Rossman3

  • 1University of Warwick, Coventry, United Kingdom.

Algorithmica
|December 5, 2022
PubMed
Summary

Researchers improved lower bounds for monotone circuit size using robust sunflowers. This work advances understanding of circuit complexity and related areas like DNF sparsification.

Keywords:
Arithmetic circuit complexityCircuit complexityComputational complexityExtremal combinatoricsMonotone arithmetic circuitsMonotone circuit complexityRobust sunflower lemmaSunflowers

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Area of Science:

  • Theoretical Computer Science
  • Combinatorics
  • Circuit Complexity

Background:

  • Robust sunflowers are a generalization of combinatorial sunflowers with applications in circuit complexity, DNF sparsification, and randomness extractors.
  • The Erdős-Rado sunflower conjecture has seen recent advancements, providing improved bounds on set systems excluding robust sunflowers.

Purpose of the Study:

  • To establish new lower bounds for the monotone circuit size of explicit n-variate monotone functions.
  • To derive a lower bound for monotone arithmetic circuit size of a related polynomial.
  • To introduce robust clique-sunflowers and prove a lower bound for the CLIQUE function's monotone circuit size.

Main Methods:

  • Leveraging the recent breakthrough on the maximum size of a w-set system that excludes a robust sunflower.
  • Developing a simple proof for the monotone arithmetic circuit lower bound.
  • Introducing and applying the concept of robust clique-sunflowers.

Main Results:

  • An improved lower bound of for the monotone circuit size of an explicit n-variate monotone function, surpassing previous bounds.
  • A lower bound of for the monotone arithmetic circuit size of a related polynomial.
  • An lower bound on the monotone circuit size of the CLIQUE function for all .

Conclusions:

  • The study significantly advances the understanding of monotone circuit complexity by establishing new lower bounds.
  • The introduction of robust clique-sunflowers provides a novel tool for analyzing specific computational problems like CLIQUE.
  • These findings have implications for DNF sparsification and the broader study of combinatorial structures in computer science.