Computational cost of matrix multiplication

Determine the exact minimal number of arithmetic operations required to multiply two n×n matrices over a field as a function of n (the computational cost of matrix multiplication), i.e., establish the optimal asymptotic complexity of matrix multiplication.

Background

Matrix multiplication is a central operation in computer science and numerical linear algebra. Strassen’s 1969 algorithm showed that the naive cubic-time method is not optimal by achieving O(n{2.805…}) operations, and subsequent work has reduced this further to O(n{2.37133…}). Despite decades of progress, the precise optimal asymptotic complexity remains unknown.

This paper studies commutative matrix multiplication schemes via flip graphs and related tensor methods, recovering known bounds for small sizes. While these results provide practical insights and tools for exploring new algorithms, resolving the overall computational cost of matrix multiplication remains a fundamental open challenge beyond the specific commutative setting considered here.

References

The computational cost of matrix multiplication is a famous open problem in algebraic complexity theory.

— Exploring Commutative Matrix Multiplication Schemes via Flip Graphs  (2506.22113 - Wood, 27 Jun 2025) in Section 1 (Introduction), first paragraph; p. 1

Despite its prominence, the computational complexity of matrix multiplication---the number of arithmetic operations needed to multiply large matrices---is unknown, and determining it is a major open question in theoretical computer science \citep{blaser2013fast}.

— Improving the matrix multiplication exponent with modern optimization and AlphaEvolve  (2608.16884 - Dupont et al., 17 Aug 2026) in Section 1, Introduction