I'm pretty sure the rat killing nature of cats is way overstated. Also, in an urban environment where they can much more easily find food thrown in trash, etc., they are unlikely to make an effort to kill rats to obtain that food. Finally, a sufficient cat population that could eradicate a rat problem would likely eradicate the bird population first.
Go read the other responder's article link. Cats can kill rats, but they generally don't bother, and are very inefficient predators of rats. Rats are just too large. Predators don't like to prey on animals that are a significant threat to them. Mice, OTOH, are easy prey for cats and they'll kill tons of them.
