Monotone Circuit Lower Bounds from Robust Sunflowers

TL;DR

This paper leverages recent advances in robust sunflowers to establish significantly stronger lower bounds on the size of monotone circuits computing certain functions, advancing understanding in circuit complexity.

Contribution

It introduces new lower bounds for monotone circuit complexity using robust sunflower techniques, improving previous bounds and extending results to related polynomials and functions.

Findings

Established an $ ext{exp}(n^{1/2-o(1)})$ lower bound for monotone circuit size of explicit functions.

Proved an $ ext{exp}( ext{Omega}(n))$ lower bound on monotone arithmetic circuit size.

Derived an $n^{ ext{Omega}(k)}$ lower bound for the CLIQUE function for $k extless n^{1/3-o(1)}$.

Abstract

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erd\H{o}s-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang gives an improved bound on the maximum size of a $w$-set system that excludes a robust sunflower. In this paper, we use this result to obtain an $\exp(n^{1/2-o(1)})$ lower bound on the monotone circuit size of an explicit $n$-variate monotone function, improving the previous best known $\exp(n^{1/3-o(1)})$ due to Andreev and Harnik and Raz. We also show an $\exp(\Omega(n))$ lower bound on the monotone arithmetic circuit size of a related polynomial. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an $n^{\Omega(k)}$ lower bound on the monotone circuit size of the CLIQUE function for all $k \le n^{1/3-o(1)}$, strengthening the bound of Alon and Boppana.