I agree that pax should keep their luggage close to them but I don't think that assigning bins would work. I agree with TMOliver, the only real way to control this is through the GAs and FAs. A gate announcement and a simple, "excuse me sir but you need to store your bag closer to your seat" would probably do the trick. Granted, it wouldn't eliminate all of the problems but it should stop most of it.

I'm curious what FAs think on this though because it obviously adds to their responsibilities and could cause problems, like Goalie said