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：我们重新审视基础的布尔矩阵乘法（BMM）问题。自50多年前代数快速矩阵乘法诞生以来，已知BMM可在真正次三次时间$O(n^\omega)$内求解（其中$\omega<3$）；大量研究致力于将$\omega$逼近2。此后，并行研究寻求具有可比速度的组合算法，但进展有限。最初的$O(n^3)$时间算法经$\log^2{n}$因子改进[Arlazarov等人; RAS'70]，随后改进为$\log^{2.25}{n}$ [Bansal和Williams; FOCS'09]，再改进为$\log^3{n}$ [Chan; SODA'15]，最终改进为$\log^4{n}$ [Yu; ICALP'15]。我们设计了一种运行时间为$n^3 / 2^{\Omega(\sqrt[7]{\log n})}$的组合BMM算法——该加速比任何多对数因子更强，且具有超越三次时间的效果。这近乎否定20世纪90年代以来关于BMM不存在真正次三次组合算法的猜想。该广泛持有的猜想是数十个细粒度困难性结论的基础。我们的主要技术贡献是提出了布尔矩阵（或等价地，二分图）的一种新的正则性分解定理，该正则性概念最近在通信复杂性背景下被引入并进行了分析性研究[Kelley, Lovett, Meka; arXiv'23]，且与近期关于无三项算术级数集合的工作中的类似概念相关[Kelley, Meka; FOCS'23]。