The fact that you are able to drive for 20 minutes on a dead battery tells me the alternator isn't the problem.

One other thought--when I have a dead battery and then jump it, I want to drive more than just 20 minutes to make sure it charges enough to be able to start it the next time. Starting the car is what takes the most juice so a short drive with a dead battery will not charge it enough. So while it sounds likely that the battery needs to be replaced, it could just be that it drained because a light was left on or something and it just hasn't been able to charge sufficiently. If you have a trickle charger I would plug it in overnight and see if it starts then.