We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic $O(n^\omega)$ time, where $\omega<3$; much work has gone into bringing $\omega$ closer to $2$. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The naive $O(n^3)$-time algorithm was initially improved by a $\log^2{n}$ factor [Arlazarov et al.; RAS'70], then by $\log^{2.25}{n}$ [Bansal and Williams; FOCS'09], then by $\log^3{n}$ [Chan; SODA'15], and finally by $\log^4{n}$ [Yu; ICALP'15]. We design a combinatorial algorithm for BMM running in time $n^3 / 2^{\Omega(\sqrt[7]{\log n})}$ -- a speed-up over cubic time that is stronger than any poly-log factor. This comes tantalizingly close to refuting the conjecture from the 90s that truly subcubic combinatorial algorithms for BMM are impossible. This popular conjecture is the basis for dozens of fine-grained hardness results. Our main technical contribution is a new regularity decomposition theorem for Boolean matrices (or equivalently, bipartite graphs) under a notion of regularity that was recently introduced and analyzed analytically in the context of communication complexity [Kelley, Lovett, Meka; arXiv'23], and is related to a similar notion from the recent work on $3$-term arithmetic progression free sets [Kelley, Meka; FOCS'23].

翻译：我们重新审视布尔矩阵乘法（BMM）这一基础问题。自五十年代代数快速矩阵乘法理论提出以来，人们已认识到BMM可在$O(n^\omega)$（其中$\omega<3$）的严格次立方时间内求解；大量研究致力于将$\omega$逼近至$2$。与此同时，另一研究方向试图构建具有可比速度的组合算法，但进展有限。朴素$O(n^3)$算法最初被改进$\log^2{n}$倍[Arlazarov等; RAS'70]，随后提升$\log^{2.25}{n}$倍[Bansal与Williams; FOCS'09]，再改进$\log^3{n}$倍[Chan; SODA'15]，最终优化$\log^4{n}$倍[Yu; ICALP'15]。我们设计了一种组合BMM算法，其运行时间为$n^3 / 2^{\Omega(\sqrt[7]{\log n})}$——该立方时间加速效果超越任何多对数因子。这一突破近乎否定了九十年代关于“严格次立方组合BMM算法不可能存在”的猜想，该流行猜想正是数十项细粒度硬度研究的理论基础。我们的核心技术贡献在于提出了一种基于新型正则性概念的布尔矩阵（或等价二部图）正则分解定理。该正则性概念近期在通信复杂性研究中被引入并解析分析[Kelley, Lovett, Meka; arXiv'23]，且与近期关于无三项算术数列集合的研究[Kelley, Meka; FOCS'23]中的类似概念相关联。