Matrix multiplication exponent equals 2

Determine whether the exponent ω of square matrix multiplication over the reals is equal to 2.

Background

Recent nearly-optimal linear programming algorithms have arithmetic complexity that depends on the matrix multiplication exponent ω. If ω were 2, these LP algorithms would achieve near-quadratic time (up to polylogarithmic factors).

The long-standing conjecture that ω = 2 remains open, and resolving it would immediately sharpen the best-known LP complexity bounds described in the paper.

References

Were the exponent of matrix multiplication ultimately shown to be 2 (as is conjectured, although progress in reducing it has slowed considerably), the complexity of the algorithms in would become \tilde{O}(n{2+1/6}) while for the algorithm of it would become O(n{2+1/18}).

— Optimization in Theory and Practice  (2510.15734 - Wright, 17 Oct 2025) in Section 4, Linear Programming, Subsection "Recent Developments in Complexity Analysis of Interior-Point Methods"

No larger lower bound is known, leading many to optimistically conjecture that ω=2.

— More Asymmetry Yields Faster Matrix Multiplication  (2404.16349 - Alman et al., 2024) in Section 1 (Introduction)

We further show that the above algorithm for $t = 2$ can be implemented in time $n{\omega + o(1)}$ for $\omega$ the constant of square matrix multiplication and succeeds when $\lambda > 0.3320$; under the folklore conjecture that $\omega = 2$, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques.

— Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-cliques  (2609.24780 - Kunisky et al., 21 Sep 2026) in Abstract; Section 1, subsection “Main results”

More precisely, the asymptotic rank of the $n\times n$ matrix multiplication tensor is $n\omega$ for some (unknown) constant $\omega$, the exponent of matrix multiplication.

— Asymptotic completions of preordered semirings  (2609.31021 - Vrana, 25 Sep 2026) in Section 1, Introduction, paragraph discussing the semiring of tensors and matrix multiplication