Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as \begin{align} B_{i,j} = \begin{cases} A_{i,j}, & i\in\mathcal{I},\\ 0, & \text{otherwise}, \end{cases} \end{align} where $\mathcal{I}$ is a subset of $\{1,2,\cdots,m\}$.

For a constant size of $\mathcal{I}$, how does one should select $\mathcal{I}$ such that the following quantity becomes maximum $$\|\mathbf{B}^\mathrm{T}\mathbf{B}\|_1+\|\mathbf{B}\mathbf{B}^\mathrm{T}\|_1,$$ where $\|.\|_1$ is defined as the sum of the all elements of the matrix, and superscript $(.)^{\mathrm{T}}$ denotes the transpose operatrion.