I know the constraints matrix $A$ of a linear program $$ \min c^Tx \text{ subject to } b\le Ax $$ is totally unimodular. So, the program has integral solutions for integral vector $b$.
Is this is also the case for the following problem: $$ \min c^Tx \text{ subject to } b_1\le Ax\le b_2 $$ where $b_1$ and $b_2$ are integral vectors and $A$ is totally unimodular. Does it have integral solutions, too?