Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, B_2)$ is maximum, where $w(A_i, B_i) := \sum\limits_{\{a,b\}\in (A\times B)\cap E}w(a,b)$.

I think this problem is already known but I couldn't find any reference. Does anyone know any related problems? Or is this problem NP-hard?