TITLE: Characterizing sums of permutation matrices
QUESTION [9 upvotes]: Given an $n$ by $n$ matrix $A$ whose rows and columns sum to $m \in \mathbb N$ and entries are nonnegative integers, does there exist a permutation matrix $P$ such that $A - P$ has only nonnegative entries?
If this is true, then we can write $A$ as the sum of $m$ permutation matrices, and I'll have a good Thanksgiving.

REPLY [2 votes]: By Birkhoff–von Neumann theorem, every doubly stochastic matrix is a convex combination of permutation matrices. Therefore $\frac1mA=\sum_{k=1}^r c_kP_k$ for some permutation matrices $P_1,P_2,\ldots,P_r$ and some $c_1,c_2,\ldots,c_n>0$. Hence $A\ge mc_1P_1$ entrywise. As $A$ is an integer matrix, it follows that $A\ge P_1$ entrywise, i.e. $A-P_1$ is entrywise nonnegative.