Sunflowers theorems in computational complexity

Author(s)

Bruno Pasqualotto Cavalar

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2020-09-09

Examining board members

Yoshiharu Kohayakawa; Mrinal Kumar; Benjamin Evan Rossman

• Advisor: Yoshiharu Kohayakawa

Abstract

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a small circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erds and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called robust sunflowers, to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossmans bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain the most significant progress on the sunflower conjecture since its inception in 1960. In this work, we will show how the recent progresses in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, thus breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. (AU)

• FAPESP's process: 18/05557-7 - Computational complexity and extremal combinatorics
• Grantee: Bruno Pasqualotto Cavalar
• Support Opportunities: Scholarships in Brazil - Master