Integer Rounding for Polymatroid and Branching Optimization Problems

Where matrix $M\geqq 0$ and vector $w\geqq 0$ have rational entries, define $r^* (w) = \max \{ 1 \cdot y:yM\leqq w,y\geqq 0 \}$, $z^* (w) = \max \{ 1 \cdot y:yM\leqq w,y\geqq 0,y\,{\text{integral}} \}$. Integer round-down holds for M if, for all integral $w\geqq 0$ , $\lfloor r^* (w) \rfloor = z^* (w)$. Similarly, when $\lceil r_* (w) \rceil = z_* (w)$ for all integral $w\geqq 0$, where $r_* (w) = \min \{ 1 \cdot y:yM\leqq w,y\geqq 0 \}$, $z_* (w) = \min \{ 1 \cdot y:yM\geqq w,y\geqq 0,y \,{\text{integral}} \}$, integer round-up holds for M. The integer round-down and round-up properties are shown to hold for certain matrices related to integral polymatroids and branchings in directed graphs.