Is there a solution for this maze problem in polynomial time?

Suppose you have a maze represented by a graph where each vertex represents a room and edges represent paths between rooms and each edge has a weight denoting the time it takes to go that way. Now here comes the tricky part: suppose each room needs a set of keys for you to enter, and inside each room you can find another set of keys. Keys can be repeated and one key may be needed to enter on more than one room. You can only enter the room if you have all keys required.

You have an arbitrarily large number of chests with gold inside and each chest needs one key which can be found in the maze. You already have a set of keys that you can use. You can use a key more than once. The question is: how do you collect the keys you need as fast as possible?

If the number of keys is unlimited, the problem is NP-hard, by a straightforward reduction from the traveling salesman problem: simply put a different key in each vertex, and have a single chest that requires all the keys.

StackExchange Computer Science Q#96729, answer score: 2