A certain instance of the Set Covering problem

Question

Is there any useful structure associated with the following instance of the Set Covering problem?

Let $G$ be a weighted graph and let $\mathcal{P}$ denote the set of all shortest paths between all pairs of nodes in $G$ whose length exceeds some threshold $r$. Construct an instance of set cover in which each element $e_i$ is associated with a path $P_i$ in $\mathcal{P}$, and each set $S_j$ is associated with a node $n_j$ in $G$, and $S_j$ contains precisely those elements $e_i$ such that $P_i$ contains $n_j$.

Does anything change if $\mathcal{P}$ consists of all paths in $G$ with length at least $r$? As Tony Huynh pointed out, if $r$ is small (so $\mathcal{P}$ consists of all paths in $G$), then this is just the vertex covering problem because an edge is a shortest path.

Answer 1

For the variant that $\mathcal{P}$ consists of all paths, the problem is equivalent to minimum vertex cover, and hence is NP-complete. To see this, I assume that single vertices do not count as paths, since otherwise you have to take all $S_v$, and the problem is uninteresting. But edges are certainly paths, so the set of vertices you pick must be a vertex cover of $G$. On the other hand, if $X$ is a vertex cover, then taking $\mathcal{S}$ to be all $S_x$ such that $x \in X$ will obviously cover all paths since $\mathcal{S}$ covers all edges.

Note that this solves the original problem if the weights satisfy the triangle inequality, since in that case, all edges are shortest paths.