New Graph Decompositions and Combinatorial Boolean Matrix Multiplication Algorithms

TL;DR

This work tackles Boolean Matrix Multiplication (BMM) from a combinatorial perspective. It introduces grid regularity and a new regularity decomposition to break arbitrary inputs into quasi-polynomially many regular pieces, enabling efficient reductions to triangle detection. By combining AB-/A-decompositions with a deterministic sifting and density-increment framework, the authors achieve a deterministic algorithm computing the Boolean product in time $n^3 / 2^{\Omega(\sqrt[7]{\log n})}$, approaching subcubic performance and providing wide-ranging corollaries for related problems. The approach builds on the grid-norm framework of Kelley–Lovett–Meka and yields near-subcubic, worst-case guarantees, offering a principled path toward breaking the enduring conjecture that truly subcubic combinatorial BMM is impossible and opening avenues for further refinements and practical structure-to-randomness techniques.

Abstract

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 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 $n^3 / 2^{Ω(\sqrt[7]{\log n})}$ -- 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].

Theorems & Definitions (27)

• Conjecture 1.1: Combinatorial BMM
• theorem 1.2: Combinatorial BMM
• theorem 2.1: Regular Matrices Have Uniform Products KelleyLM23
• theorem 3.1: $AB$-Decomposition
• theorem 3.2: Sifting
• theorem 3.3: $A$-Decomposition
• lemma 4.1: Inverse of Markov's Inequality
• lemma 4.2: Recursive Sifting
• lemma 4.3: Deterministic Regularity Approximation
• theorem 4.3: Sifting