Let $x_{ij}$ where $i,j\in [n]$ be variables. Consider the polyhedron defined by:
$x_{ii}=0$, $x_{ij}\geq 0$. For each $i$, $\sum_{j=1}^nx_{ij}=\sum_{j=1}^nx_{ji}=y_i$ for some fixed constant $y_i>0$.
Provided that the polyhedron is non-empty, what are the vertices of this polyhedron?
Another way to describe the polyhedron is that we have a set of $n\times n$ square matrix $X$, where the diagonals of $X$ are zero, each entry of $X$ is non-negative, and the sum of $i$-th row and $i$-th column both equal to a constant $y_i>0$.
For $y_1=y_2=\cdots=y_n=y$, the vertices are the set where $x_{ij}=y$ for $(i,j)\in S$ and $x_{ij}=0$ for $(i,j)\notin S$. Here $S$ is an index set such that:
(a) $S$ contains $n$ elements;
(b) for any $(i,j), (i',j')\in S$, if $(i,j)\neq(i',j')$ then $i\neq i'$ and $j\neq j'$; (In some sense (a) and (b) describes the set of possible positions for each number in a Sudoku)
(c) $(i,i)\notin S$ for any $i\in [n]$.
However, I am stuck on the general case. Any thoughts?
Edit: BTW I would also be interested to know when this polyhedron is empty.