Finding a dense submatrix of a random matrix. Sharp bounds for online algorithms

We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences \cites{madeira-survey,shabalin2009finding} and is known to exhibit a computation-to-optimization gap between the optimal value and best values achievable by existing polynomial time algorithms. In this paper we consider the class of online algorithms, which includes the best known algorithm for this problem, and derive a tight approximation factor $\frac{4}{3\sqrt{2}}$ for this class. The result is established using a simple implementation of recently developed Branching-Overlap-Gap-Property \cite{huang2025tight}. We further extend our results to $({\mathbb{R}}^n{)}^{\otimes p}$ tensors with i.i.d. Gaussian entries, for which the approximation factor is proven to be $2\sqrt{p}\mathrm{/}(1+p)$.