Suppose you have a weighted connected graph, $G(V, E)$, with $n$ nodes such that every node has a edge to every other node (a large clique). You are also given a set of sets, $\{l_1, l_2, ... l_n\}$ so that every $l_i$ corresponds to $v_i \in V$.
Starting from any node $s \in V$, find the shortest route $G^\prime(V^\prime, E^\prime)$ so that for every corresponding $l$ and $l^\prime$, $\bigcup_{i=1}^{n}l_{i} = \bigcup_{j=1}^{k}l_{j}^\prime$, and it terminates on $s$.
In case my rather poor notation is insufficient in getting the idea across, imagine you had a shopping list of items and a bunch of stores to choose from. You can travel from any store to any store. Not every store will have every item you need and there will be some overlap between stores. Find the shortest route to pick up all the items you need and return home.