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Hard Submatrices for Non-Negative Rank and Communication Complexity

Author Pavel Hrubeš

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Author Details

Pavel Hrubeš
  • Institute of Mathematics of ASCR, Prague, Czech Republic

Funding

  • Hrubeš, Pavel: This work was supported by Czech Science Foundation GAČR grant 19-27871X.

Acknowledgements

The author wants to thank Anup Rao for useful comments.

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Pavel Hrubeš. Hard Submatrices for Non-Negative Rank and Communication Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.13

Abstract

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 r³ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Non-negative rank
  • communication complexity
  • extension complexity

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