You approach a bridge and see a troll. He throws down finite $n>0$ bags. Each bag, $B_i$, contain 1 or more of 4 possible buttons. $B_i$ contains $S_i \subseteq \{green, red, blue, yellow\} \land S_i \neq \emptyset$.
The contents of each bag is written on the bag - e.g. bag $B_i$ may have $\{red, green\}$ written on it.
The troll says, I know buttons are valuable in your neighbourhood and you can't get past me until you pick a set of bags such that you minimise $\frac{colours}{bags}$.
What is the best way to find the set of bags you should take?
Bonus points if you can tell me what this (NP?) problem is called.