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The Log-Rank Conjecture: New Equivalent Formulations

Authors Lianna Hambardzumyan , Shachar Lovett , Morgan Shirley

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LIPIcs.CCC.2026.7.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • cross-intersecting set systems
  • Log-rank conjecture
  • monochromatic rectangle
  • partition number

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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 ± 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.

Cite As Get BibTex

Lianna Hambardzumyan, Shachar Lovett, and Morgan Shirley. The Log-Rank Conjecture: New Equivalent Formulations. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 7:1-7:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.7

Author Details

Lianna Hambardzumyan
  • The University of Copenhagen, Denmark
Shachar Lovett
  • University of California San Diego, CA, USA
Morgan Shirley
  • Lund University, Sweden

Funding

Most of the work was done while the first and third authors were postdoctoral researchers at University of Victoria, Canada, and funded by NSERC.
  • Hambardzumyan, Lianna: This work is funded by the European Research Council (ERC) under grant agreement no. 101125652 (ALBA).
  • Lovett, Shachar: Research supported by Simons Investigator Award #929894 and NSF award CCF-2425349.
  • Shirley, Morgan: Supported by Knut and Alice Wallenberg grant KAW 2023.0116.

Acknowledgements

We are grateful to Adi Shraibman and Amir Yehudayoff for enlightening discussions.

References

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