Boolean-OMv to counting-OMv reductions

Determine whether a truly subcubic-time algorithm for Boolean Online Matrix-Vector multiplication (Boolean-OMv) implies truly subcubic-time OMv algorithms for the counting variants of equality and dominance matrix-vector products, specifically for Counting-Equality-OMv defined by u[i] := |{k ∈ [n] : M[i,k] = v[k]}| and Counting-Dominance-OMv defined by u[i] := |{k ∈ [n] : M[i,k] ≤ v[k]}|, where M ∈ ℤ^{n×n} is given for preprocessing and queries provide v ∈ ℤ^n online.

Background

The paper establishes fine-grained equivalences between multiple OMv variants that do not involve counting (Boolean, ∃Equality, ∃Dominance, Min-Witness, Min-Max, and Bounded Monotone Min-Plus), showing that they either all admit truly subcubic algorithms or none of them do.

The authors explicitly note that their equivalences cover non-counting variants only and raise the question of whether similar implications carry over to counting outputs, which are often harder due to the need to aggregate counts rather than make existential decisions. They also connect this to broader work on counting-to-decision reductions in fine-grained complexity.