Non-Boolean OMv: One More Reason to Believe Lower Bounds for Dynamic Problems

TL;DR

This paper introduces new hypotheses based on variants of matrix multiplication that are believed to be harder than Boolean matrix multiplication, and proves their equivalence to the standard OMv Hypothesis, strengthening the foundation of dynamic problem lower bounds.

Contribution

It establishes the first fine-grained equivalence class for dynamic problems by showing the equivalence of OMv with variants based on different matrix products.

Findings

New hypotheses for matrix product variants are introduced.

Proved equivalence between these variants and the standard OMv Hypothesis.

Strengthens the theoretical basis for lower bounds in dynamic problems.

Abstract

Most of the known tight lower bounds for dynamic problems are based on the Online Boolean Matrix-Vector Multiplication (OMv) Hypothesis, which is not as well studied and understood as some more popular hypotheses in fine-grained complexity. It would be desirable to base hardness of dynamic problems on a more believable hypothesis. We propose analogues of the OMv Hypothesis for variants of matrix multiplication that are known to be harder than Boolean product in the offline setting, namely: equality, dominance, min-witness, min-max, and bounded monotone min-plus products. These hypotheses are a priori weaker assumptions than the standard (Boolean) OMv Hypothesis. Somewhat surprisingly, we show that they are actually equivalent to it. This establishes the first such fine-grained equivalence class for dynamic problems.