Tight complexity for intermediate matrix product variants in the static setting

Ascertain whether the optimal offline time complexity for intermediate matrix multiplication variants—including min-max product, min-witness product, equality (Hamming) product, dominance product, threshold product, plus-max product, and ℓ_{2p+1} product—necessarily matches the current function-of-ω bounds that converge to O(n^{2.5}) when the matrix multiplication exponent ω=2, i.e., determine if these bounds are tight for all such problems or if faster algorithms exist.

Background

The introduction surveys numerous static matrix product variants believed to be harder than Boolean product but easier than min-plus/APSP. For these problems, the best known algorithms have running times parameterized by the matrix multiplication exponent ω, approaching O(n{2.5}) if ω=2.

Despite partial tight reductions suggesting these bounds might be optimal, the authors explicitly note that it remains unknown whether this is indeed the correct complexity across all listed variants, framing a broad open question about optimality of current upper bounds.

References

The fastest known algorithms for these problems have running times that are functions of the matrix multiplication exponent \omega, and they converge to O(n{2.5}) when \omega=2. Although it is still an open problem whether this is necessarily the right complexity for all these problems, there are some partial results in the form of tight fine-grained reductions that suggest it might be the case.

— Non-Boolean OMv: One More Reason to Believe Lower Bounds for Dynamic Problems (2409.15970 - Hu et al., 24 Sep 2024) in Section 1 (Introduction)