Given parameters $r < n$, define $m = {n \choose r}$ and let $A$ be the $n$ by $m$ matrix whose columns are all the vectors with $r$ $1$'s and $n-r$ $0$'s. Let $b$ be a positive $n$-vector. Is there a simple solution to the linear program:

$$ \max \sum_i x_i$$

subject to

$$ Ax \le b$$

and

$$ x\ge 0$$

?