The Log-Rank Conjecture: New Equivalent Formulations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911472624336896 |
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| author | Hambardzumyan, Lianna Lovett, Shachar Shirley, Morgan |
| author_facet | Hambardzumyan, Lianna Lovett, Shachar Shirley, Morgan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Log-Rank Conjecture: New Equivalent Formulations Hambardzumyan, Lianna Lovett, Shachar Shirley, Morgan Computational Complexity Combinatorics 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. |
| title | The Log-Rank Conjecture: New Equivalent Formulations |
| topic | Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2510.02583 |