How do I permanently get rid of fruit flies and gnats?

Fruit flies and gnats are flying around my house all the time. I don't leave any food out. Don't throw away any food in the indoor trash. I rinse all my dishes and put them right in the dishwasher. I don't understand where they are coming from and I want them gone. I don't want to lure them to vinegar or trap them, I just want them all dead or gone and not to come back. I have to be careful about pesticides because I have a cat and a dog indoors, and I can't have candles with flames because the cat thinks they are fun to play with and keeps burning her whiskers off. Any ideas?