its all in the math... first you have to know what the supplied air flow to the room is. Then you can calculate the needed exit area under the door (h x w).... but I think just about everyone here will agree that normally you would have to under cut the door by about 2 to 3 or more inches for the air flow math to work out correctly.
The alternatives are:
1. install return ductwork in each room.
2. Install transfer ducts (sometimes called jump ducts) from each room to open main areas (your hall way if I read correctly)
3. install a grill in each door, of the size needed for the airflow into the room.
From experiance on my home, transfer ducts made a huge difference!!!!! kids sleep with their doors closed and putting in the transfer ducts keeps the rooms at the same temp as the rest of the house and the hvac system is much more efficent, which has more then paid for the diy cost of the installation!!!! The ducts basically allow the air from the "sealed" off rooms to get back to the air handler. again, a huge difference in comfort and economy !!!!