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

2024-09-24

arXiv.org

AI Summary

The Online Boolean Matrix-Vector Multiplication (OMv) hypothesis—widely used to establish conditional lower bounds for dynamic problems—lacks strong credibility due to its reliance on the unresolved hardness of Boolean matrix multiplication. Method: This paper introduces and systematically studies five non-Boolean OMv variants—Equality, Dominance, Minimum Witness, Min-Max, and Bounded Monotone Min-Plus Product—whose offline counterparts are known to be strictly harder than Boolean matrix multiplication. Through fine-grained reductions, the authors rigorously prove that each variant is computationally equivalent to the standard OMv hypothesis under fine-grained complexity assumptions. Contribution/Results: This is the first work to establish fine-grained equivalence between multiple non-Boolean online query hypotheses and OMv. It strengthens the theoretical foundation of dynamic problem lower bounds and, for the first time, constructs a fine-grained equivalence class in dynamic computational complexity—providing a richer, more robust set of assumptions for future conditional lower bound research.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Proposing harder matrix multiplication variants for dynamic lower bounds
• Establishing equivalence between new hypotheses and Boolean OMv
• Providing more believable assumptions for dynamic problem hardness

Innovation

• Proposes non-Boolean OMv variants for dynamic problems
• Shows equivalence between Boolean and non-Boolean hypotheses
• Establishes fine-grained equivalence class for dynamics