Hard submatrices for non-negative rank and communication complexity

Given a non-negative real matrix M of non-negative rank at least r , can we witness this fact by a small submatrix of M ? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: An m × n matrix of non-negative rank r...

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Bibliographic Details
Published inComputational complexity Vol. 34; no. 2; p. 9
Main Author Hrubeš, Pavel
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2025
Springer Nature B.V
Subjects
Algorithm Analysis and Problem Complexity
Algorithms
Boolean
Communication
Complexity
Computational Mathematics and Numerical Analysis
Computer Science
Linear algebra
Tightness
94A05
extension complexity
Non-negative rank
communication complexity
51M20
Online AccessGet full text
ISSN1016-3328
1420-8954
1420-8954
DOI10.1007/s00037-025-00269-4

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Summary:Given a non-negative real matrix M of non-negative rank at least r , can we witness this fact by a small submatrix of M ? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: An m × n matrix of non-negative rank r always contains a submatrix with at most r 3 rows and columns with non-negative rank at least Ω ( r log n log m ) . A similar result is proved for the 1-partition number of a Boolean matrix and, consequently, also for its two-player deterministic communication complexity. Tightness of the latter estimate is closely related to the log-rank conjecture of Lovász and Saks.
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ISSN:1016-3328
1420-8954
1420-8954
DOI:10.1007/s00037-025-00269-4