The Log-Rank Conjecture: New Equivalent Formulations

Lianna Hambardzumyan; Shachar Lovett; Morgan Shirley

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arXiv · 2510.02583

The Log-Rank Conjecture: New Equivalent Formulations

Lianna Hambardzumyan·Shachar Lovett·Morgan Shirley

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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 $\pm 1$-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.

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BibTeX RIS

Lianna Hambardzumyan, Shachar Lovett, Morgan Shirley. 2025-10-02. The Log-Rank Conjecture: New Equivalent Formulations. https://arxiv.org/abs/2510.02583

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