Maximizing sum of ranges of vertices edges
I'll start by giving a negative example to @seteropere algorithm:

Let $G=(V,E)$, where $$V=\{x_1,x_2,x_3,y_1,y_2,y_3\}\ \ \ ,\ \ \ E=\{(x_i,y_i)\mid i\in [3]\}\cup\{(x_1,x_2),(x_1,x_3)\}$$
Let the weight collection be $\{3,3,2,1,1\}$.

The proposed algorithm starts by assigning weights $1,2$ and $3$ for $x_1$, and assume that $(x_1,y_1)$ was assigned 1.

This already means that the optimal solution profit (5, see figure) is not obtainable.







  That said, this problem is poly time solvable after all , greedily,
  but differently.


Hint: given a graph $G$ such that $\Delta(G)\geq 2$, prove that in the optimal solution there exists a vertex $v$ such that the maximal number and the minimal number are both in $w(v)$.



P.S. 

The opposite problem, in which we aim to minimize $$\sum_{v \in V}{max(w(v)) - min(w(v))}$$ 

Is quite interesting by itself.