My pool is fully screened in, so it gets less debris from the environment. A few weeks ago, I decided to try running the pump only half an hour per day starting when I add chlorine.
I've invested in a nice skimming net, and spend a few minutes daily skimming, and vacuum any debris from the bottom of the pool on the weekend, which seems to be a little more than normal now, but not excessive. The pool is clear and looks pretty good, with the trade-off of being slightly less ascetic with a little bit of visible debris now, especially before skimming.
A short time after I started that, I tested for CYA for the beginning of the season, and saw that lo and behold, I actually needed to raise the level, so the floating chlorinator and trichlor tablets that the pool store sold me when I was ignorant went back in the pool to stay until the desired level is attained (hey, I already paid for them)!
Good idea or bad, I've actually been experimenting going a day or two before running the pump at all with the floating chlorinator in the pool. I've seen nothing obvious bad from this experiment, but I am wondering if the pump still needs to run for a short period every day to mix the chlorine, even if it was introduced by a chlorinator that floats from place to place.