The Log-Rank Conjecture: New Equivalent Formulations

2025-10-02

AI Summary

The Log-Rank Conjecture—a central open problem in complexity theory concerning the relationship between the rank and the partition number of Boolean matrices—remains unresolved. This paper introduces *sign-rank rectangle*, a new matrix parameter shown to be equivalent to matrix rank up to logarithmic factors. This equivalence yields a clean reformulation of the conjecture: whether any sign decomposition of a Boolean matrix can be converted into a nonnegative decomposition with quasi-polynomial overhead. The sign-rank rectangle naturally lies between rank and partition number, and—crucially—establishes, for the first time, an equivalence between the Log-Rank Conjecture and Lovett’s conjecture as well as the Singer–Sudan conjecture on cross-intersecting set systems. Methodologically, the work integrates linear algebraic techniques, combinatorial rectangle covering arguments, and tensor lifting. All results extend seamlessly to higher-order tensors, providing a unified framework for generalizing the conjecture beyond matrices.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography

Abstract

The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are quasi-polynomially related. We propose a relaxed but still equivalent version of the conjecture based on a new matrix parameter, signed rectangle rank: the minimum number of all-1 rectangles needed to express the Boolean matrix as a $pm 1$-sum. Signed rectangle rank lies between rank and partition number, and our main result shows that it is in fact equivalent to rank up to a logarithmic factor. Additionally, we extend the main result to tensors. This reframes the log-rank conjecture as: can every signed decomposition of a Boolean matrix be made positive with only quasi-polynomial blowup? As an application, we prove an equivalence between the log-rank conjecture and a conjecture of Lovett and Singer-Sudan on cross-intersecting set systems.

Problem

• The paper explores equivalent formulations of the log-rank conjecture in complexity theory
• It investigates quasi-polynomial relationships between rank and partitioning number of Boolean matrices
• The study connects the log-rank conjecture to cross-intersecting set systems conjectures

Innovation

• Introduces signed rectangle rank parameter
• Shows equivalence to rank with logarithmic factor
• Extends main result to tensor formulations