Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize  $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb R_{\geq0,\leq b}^{n\times n}\}?$$

What is a good upper bound to $$\max_{M\in\Bbb Z_{\geq0,\leq b}^{n\times n},\mathsf{rank}(M)=r}\min_{Q\in\mathscr{D}[M,b]}\mathsf{rank}(Q)?$$