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Boolean-OMv to standard integer product OMv

Determine whether a truly subcubic-time algorithm for Boolean Online Matrix-Vector multiplication (Boolean-OMv) implies a truly subcubic-time OMv algorithm for standard integer matrix-vector multiplication (Integer-Product-OMv), defined by u[i] := Σ_{k∈[n]} M[i,k] · v[k], where M ∈ ℤ^{n×n} is given for preprocessing and queries provide v ∈ ℤ^n online.

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Background

Beyond counting variants, the authors ask whether Boolean-OMv would also imply subcubic algorithms for the standard integer product in the online setting, which involves arithmetic aggregation rather than Boolean or existential operations.

They note that existing counting-to-decision reductions rely on fast algebraic integer matrix multiplication for static problems, making it unclear whether those techniques extend to online OMv settings.

References

In this paper we manage to reduce to Boolean-OMv from OMv variants that do not involve counting. We leave it open whether a subcubic algorithm for Boolean-OMv would imply subcubic OMv algorithms for, e.g., the counting variants of the equality and dominance products (i.e., u[i] := #{k \mid M[i,k] = v[k]}, and u[i] := #{k \mid M[i,k] \leqslant v[k]}, respectively), or at least for the standard integer product (u[i] := \sum_k M[i,k] \cdot v[k]).

— Non-Boolean OMv: One More Reason to Believe Lower Bounds for Dynamic Problems (2409.15970 - Hu et al., 24 Sep 2024) in Subsection “Open problems”