Monotone Circuit Lower Bounds from Robust Sunflowers

Abstract

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity [14], DNF sparsification [6], randomness extractors [8], and recent advances on the Erdős-Rado sunflower conjecture [3, 9, 12]. The recent breakthrough of Alweiss, Lovett, Wu and Zhang [3] 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 record $\exp (n^{1/3-o(1)})$ of Harnik and Raz [7]. We also show an $\exp (\varOmega (n))$ 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 $n^{\varOmega (k)}$ 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 [1].