• DOI:10.1007/978-3-030-61792-9_25
• Corpus ID: 210837114

Monotone Circuit Lower Bounds from Robust Sunflowers

@article{Cavalar2020MonotoneCL,
  title={Monotone Circuit Lower Bounds from Robust Sunflowers},
  author={B. P. Cavalar and Mrinal Kumar and Benjamin Rossman},
  journal={Electron. Colloquium Comput. Complex.},
  year={2020},
  volume={27},
  pages={181}
}

• B. P. Cavalar, Mrinal Kumar, Benjamin Rossman
• Published 7 December 2020
• Mathematics, Computer Science
• Electron. Colloquium Comput. Complex.

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-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.