Let $\textbf{A}$ be a sparse matrix. Is there any way we can identify the index, I, such that a re-indexing of the matrix (column and row) using I reduces the distance of off-diagonal non-zero elements from diagonal element?.
For e.g., Let $\textbf{A}=\begin{bmatrix}1&0&6\\0&2&0\\5&0&3\end{bmatrix}$. Then re-indexing the matrix with index, $I=\begin{bmatrix}1&3&2\end{bmatrix}$ gives $\textbf{A}(I,I)=\begin{bmatrix}1&6&0\\5&3&0\\0&0&2\end{bmatrix}$
PS: The minimal distance condition on one side (upper or lower triangular) is sufficient. i.e., if we are able to re-index the matrix so that off-diagonal elements on one side is close to diagonals.
Finding optimal solution is NP-hard problem, therefore approximate method must be used. General method is to find the reverse Cuthill-McKee ordering of $A + A^T$, $AA^T$ or $\left[\begin{array}{cc}0 & A\\ A^T & 0\end{array}\right]$.
For unsymmetric matrices such problem is analysed in Reid, J. K., and J. A. Scott. "Reducing the total bandwidth of a sparse unsymmetric matrix." SIAM journal on matrix analysis and applications 28.3 (2006): 805-821.