Reduction to a vertex cover problem-like with weighted vertices and edges Description

Let us define a new problem with an instance $I = (G = (V, E), K, L)$, whereas:


$G$ is an undirected graph
$K \le |V|$
$L &gt; 0$ is the maximum limit
Each vertex $v \in V$ has a weight $W(v)$ 
Each edge $e \in E$ has a length $W(e)$


Let $P(v)$ be a function that returns the minimum length of a path (basically shortest path) from vertex $v$ to a vertex in $V'$.

The decision question is whether there exists a vertex set $V' \subseteq V, |V'| \le K$, such that:

$$
\sum_{v \in V} W(v) \cdot P(v) \le L
$$



Example

Consider the following graph $G$, with $K = 1$ and $L = 9$:



Taking the set $\{v_3\}$ as $V'$ would be the solution to the question, because the total cost is:

$$
3 \cdot 0 + 2 \cdot 2 + 1 \cdot 5 = 9
$$

Therefore, this is a yes-instance.



Question

How do I prove that this problem is in $\mathsf{NPC}$? I tried reducing a $\text{VC}$-instance to this, but that does not seem to work.

What I have tried as well is by converting the above undirected graph to a directed graph with the weighted paths already computed (this is polynomial computable):

$$
\begin{array}{l|l|l}
        &amp; v_1 &amp; v_2 &amp; v_3 \\
\hline
    v_1 &amp; 0   &amp; 3   &amp; 5   \\
\hline
    v_2 &amp; 6   &amp; 0   &amp; 4   \\
\hline
    v_3 &amp; 15  &amp; 6   &amp; 0
\end{array}
$$

However, I'm not entirely sure what $\mathsf{NPC}$ problem to reduce from. I got the tip to use the $\text{CLIQUE}$ problem from this answer, but I do not see how to perform the reduction, so that seems unlikely. To me, this looks more like something that can be reduced from the $\text{KNAPSACK}$-problem or the $\text{SET COVER}$-problem, using the weights from the transformed directed graph, but I fail to see how exactly to this. I am starting to think that the transformed directed graph is of no use here.

What $\mathsf{NPC}$-problem can I use for the reduction?
This problem is a generalization of Dominating Set, and if you let weights of vertices and edges be 1, then you have exactly Dominating Set.

The problem definition of Dominating Set is:


  Given a graph $G = (V, E)$ and an integer $k$, is there a set $V' \subseteq V$ of size at most $k$ such that the distance of any vertex $v \in V \setminus V'$  to $V'$ is 1.


(This is assuming that the distance from $v' \in V'$ to $V'$ is defined as 0.)

It follows that WVE is NP-hard.  This is occasionally called proof by generalization.