Hard submatrices for non-negative rank and communication complexity

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00638437" target="_blank" >RIV/67985840:_____/25:00638437 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s00037-025-00269-4" target="_blank" >https://doi.org/10.1007/s00037-025-00269-4</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00037-025-00269-4" target="_blank" >10.1007/s00037-025-00269-4</a>

Result language
angličtina
Original language name
Hard submatrices for non-negative rank and communication complexity
Original language description
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(rlognlogm). 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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA25-16311S" target="_blank" >GA25-16311S: Complexity of computation, communication, and search</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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