For example, if we have a graph G = (V, E) and a subset of vertices $U \subset V$. We can set $w(e)$ where $e \in E$ to be a non-negative real number. We want to minimize the total edge weight, but make sure that for all $(u, v) \in U \times U$, $l(u, v) \geq c$, where $l(u, v)$ is the shortest path from u to v, and c is a given constant.
I suspect that this problem is NP hard, could someone come up with a polynomial solution to this?