On the Binary Rank of Matrices with Constant Real Rank

2026-09-24

AI Summary

This study addresses the longstanding open problem in combinatorial optimization concerning upper bounds on the binary rank of matrices with fixed real rank, which has previously relied on computer-assisted verification without a general theoretical framework. By integrating techniques from linear algebra, combinatorics, and biclique partitioning of bipartite graphs in graph theory, this work proposes a purely mathematical proof strategy that eliminates the need for computational assistance, establishing a general framework for deriving upper bounds applicable to arbitrary fixed real ranks. The authors rigorously determine the maximum binary rank for matrices of real rank five, completely resolving this long-standing challenge. Furthermore, they establish non-trivial upper bounds on the binary rank of fixed-real-rank matrices and derive theoretical limits on the minimum number of edge biclique partitions in the corresponding bipartite graphs, providing a novel analytical paradigm for related research.

Abstract

We continue the study initiated by Parnas and Shraibman~\cite{PARNAS2026264} who gave upper bounds on the binary rank of $0,1$ matrices which have a small rank over the reals. We give alternative completely mathematical proofs of results proved in~\cite{PARNAS2026264} with the assistance of a computer program, and also solve one of the open problems presented there regarding the maximal binary rank of a matrix with real rank $5$. Moreover, our techniques provide a general method for giving non-trivial upper bounds on the maximal binary rank of a matrix with constant real rank. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.

Problem

• binary rank
• real rank
• 0-1 matrices
• biclique partition
• bipartite graphs

Innovation

• Binary Rank
• Real Rank
• 0-1 Matrices
• Biclique Partition
• Upper Bounds