Online Boolean matrix-vector multiplication conjecture

Prove or refute the conjecture that, for every constant δ>0, no randomized algorithm solves Online Boolean Matrix-Vector Multiplication for an n×n Boolean matrix and n-dimensional vectors in total time O(n^{3−δ}) with success probability at least 2/3.

Background

The paper uses the Online Boolean Matrix-Vector Multiplication (OMv) conjecture as the fine-grained complexity assumption underlying its conditional lower bound for dynamic stationary-distribution queries on rapidly mixing directed graphs. The conjecture rules out substantially subcubic total time for processing online Boolean matrix-vector products.

Under this conjecture, the paper derives that no randomized fully dynamic data structure can simultaneously achieve strongly sub-square-root update and query times for approximate stationary-distribution queries on certain rapidly mixing directed multigraphs.

References

For every constant $\delta>0$, no randomized algorithm solves Online Boolean Matrix-Vector Multiplication (OMv) for a boolean matrix of size $n \times n$ and vectors of size $n$ in total time $O(n{3-\delta})$ with probability at least $2/3$.

— Fully Dynamic Random-Walk Quantities on Undirected Expanders and $κ$-Uniform Directed Graphs  (2610.17712 - Peng et al., 13 Oct 2026) in Conjecture 2.5, Section 2, Preliminaries

There is no randomized algorithm for OMv with success probability at least \frac 1 2, O(\poly(n)) preprocessing time and O(n{3 - \varepsilon}) computation time for any \varepsilon > 0.

— Incremental Exact Vertex Connectivity  (2610.21531 - Adena et al., 15 Oct 2026) in Section 2, “Lower Bound Overview,” Conjecture (OMv problem)