I have had this situation as well. Me window, middle passenger having to sit partly on my seat (with arm rests up) and the fat person on the aisle - who took over half the middle seat.

My opinion - if you can not sit in the seat with the arm rests down - then you need to buy two seats. I think Southwest is right on with their policy. If the fat person does not buy another seat, and does not fit in one seat, then they should be removed from the flight.