I have learned that polymatroids are of the form $$\left\lbrace x\in \Bbb R^N_{\ge 0}\mid \sum_{i\in A}^n x_i\le p(A), \forall A \subseteq \lbrace 1,\ldots,N \rbrace\right\rbrace$$ where $p$ is is non decreasing and submodular. I also learned that the extreme points can be determined as
$$x_{\pi(1)}=p({\pi(1)})$$
$$x_{\pi(k)}=p(\lbrace{\pi(1)},\ldots,\pi(k)\rbrace) - p(\lbrace{\pi(1)},\ldots,\pi(k-1)\rbrace)$$
where $\pi$ is a permutation of the elements $\{1,\ldots,N\}$. I also learned that solving a linear program over a polymatroid involves ordering the elements in the cost vector. Can someone provide an illustrative example to these concepts? I having trouble understanding how I could actually apply these concepts to a problem. I would think a real world word problem would be most helpful. Please demonstrate the greedy algorithm. Thanks for any additional insights.