New Graph Decompositions and Combinatorial Boolean Matrix Multiplication Algorithms

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^ω) time, where ω<3; much work has gone into bringing ω closer to 2. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The na'ive O(n³)-time algorithm was initially improved by a log²n factor [Arlazarov et al.; RAS'70], then by log^2.25n [Bansal and Williams; FOCS'09], then by log³n [Chan; SODA'15], and finally by log⁴n [Yu; ICALP'15]. We design a combinatorial algorithm for BMM running in time n³ / 2^{ω((logn)1/7)} - 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; STOC'24], and is related to a similar notion from the recent work on 3-term arithmetic progression free sets [Kelley, Meka; FOCS'23].