The Log-Rank Conjecture: New Equivalent Formulations
Lianna Hambardzumyan, Shachar Lovett, Morgan Shirley
TL;DR
The paper addresses the log-rank conjecture by introducing signed rectangle rank $srr(M)$ as a bridge between rank and partitioning number. It proves the central bound $srr(M) \le O(r \log r)$, which yields the reformulation $\log p(M) \le (\log srr(M))^{O(1)}$ and extends the framework to tensors. It also connects the conjecture to a cross-intersecting set-systems conjecture NW, establishing an equivalence that ties a core complexity question to combinatorial set systems. Overall, the work reframes the log-rank problem as converting signed decompositions into positive ones with quasi-polynomial blowup and opens new avenues via tensor generalizations and cross-intersecting extremal questions.
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.