We have two cats, a 13 yr old and a 2.5 yr old. They have lived in harmony with Gypsy their entire lives. They used to snuggle together to sleep. In fact, the old guy has outlived two other dogs!
Then Jazzy moved in. Two years later, it's STILL a battle. We worked hard –HARD -- on "Leave it!!", and Jazzy has EXCELLENT obedience for that , except where the cats are concerned. IF I see the cat first and say, "Leave it", she almost always will. But if the cat catches her eye first, she's gone before I even know what happened, and she won't stop once she's in run-mode.
We keep a gate across the hall that leads to the kids' bedrooms and bathroom. The litter boxes, which used to be in the garage, are now in that bathroom {I HATE that!!}, as is the cat food/water.
The gate is removed at night after Jazzy is tucked into her kennel and replaced in the morning.
The cats have learned to stay in the bedrooms during the day, and roam the house at night. They don't seem to mind; all they really used to do was sleep on the couch all day. Now they sleep on the beds all day. Sometimes the younger cat comes down while Jazzy is sleeping and gets up on the back of the couch because she knows she is safe as long as she stays there {Jazzy is not allowed on the furniture and honestly never breaks that rule, even when the cat is there}.
I feel bad about it, but what else could we do? Ah well. maybe it's a more natural lifestyle{?}
Some dogs just don't get along with cats. And some cats just can't get along with dogs. Sometimes you just have to take personality into account and make accomodations.