Matrix rounding with fixed margins ($b_{ij}\in \{\lfloor a_{ij}\rfloor ,\lceil a_{ij}\rceil \}$)

A matrix of nonnegative rational numbers with integer row and column sums can be rounded entrywise up or down while preserving all those sums. Let $p_{ij}=\lfloor a_{ij}\rfloor$ and $f_{ij}=a_{ij}-p_{ij}$ be its floor and fractional part. The residual row and column sums $r_i={\sum }_j{f}_{ij}$ and $c_j={\sum }_i{f}_{ij}$ are nonnegative integers. Build a flow network with source-to-row flow network edge capacities $r_i$, row-to-column flow network edge capacities one where $f_{ij}>0$, and column-to-sink flow network edge capacities $c_j$. The fractional parts supply a feasible flow saturating all source arcs. The integral max-flow theorem supplies an integral one of the same total value. If its middle-arc flows are $h_{ij}\in \{0,1\}$, with $h_{ij}=0$ where $f_{ij}=0$, then $b_{ij}=p_{ij}+h_{ij}$ is the required matrix. This is a constructive polynomial-time algorithm; no multiplication by a common denominator is required.