Implications from Boolean-OMv to counting and integer OMv variants

Determine whether the existence of a truly subcubic-time algorithm O(n^{3−ε}) for Online Boolean Matrix-Vector Multiplication (Boolean-OMv) implies truly subcubic online algorithms for the counting versions of Equality-OMv and Dominance-OMv, specifically computing u[i] = # {k | M[i,k] = v[k]} and u[i] = # {k | M[i,k] ≤ v[k]} for each query, or at least implies a truly subcubic online algorithm for standard integer matrix–vector multiplication computing u[i] = Σ_k M[i,k] · v[k].

Background

The paper establishes fine-grained equivalences between Boolean-OMv and several non-Boolean OMv variants (equality, dominance, min-witness, min-max, and bounded monotone min-plus), thereby forming an equivalence class for dynamic problems analogous to APSP. All reductions handled in the paper concern decision-type outputs rather than counting outputs.

The authors explicitly highlight that their results do not address counting outputs or standard integer matrix–vector multiplication in the online setting. They pose whether tight implications from Boolean-OMv extend to the counting variants of equality and dominance, and even to integer matrix–vector multiplication, connecting this question to broader efforts on counting-to-decision reductions in fine-grained complexity.