2026-07-23T16:50:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27049 2026-07-23T11:36:18Z ddc:004 open_access

The Log-Rank Conjecture: New Equivalent Formulations Hambardzumyan, Lianna Lovett, Shachar Shirley, Morgan cross-intersecting set systems Log-rank conjecture monochromatic rectangle partition number 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Lianna Hambardzumyan and Shachar Lovett and Morgan Shirley 2026 Is Part Of LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2026.7 urn:nbn:de:0030-drops-270495 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.7 eng https://creativecommons.org/licenses/by/4.0/legalcode