Keeping a pet cat, without training or encouraging it to kill rats

Let's say a lay Buddhist has a rat infestation problem in his or her house. Catching rats using traps and killing them is clearly breaking the first precept of not taking a life. What about keeping a pet cat that is sufficiently well fed and taken care of, in the house, without training or encouraging it to catch rats, but nevertheless accepting the possibility that the cat may or may not choose to catch and kill rats, on its own volition, that it may find in the house? **If the pet cat does catch and kill rats, does this break the first precept?** Also, introducing a pet cat, may cause the rats to flee the house on their own, as they may find that it has become an unsuitable place to live in.