First and foremost, keep your living space as food free as possible. This will help tremendously. Also, making it harder to get into dry food like cereal; use food containers. And for food, like fruit or vegetables, that sit on the counter you should only keep the essentials out but in a tied or zip plastic bag; oranges and apples should stay in the fridge anyways, and bananas will stay longer in a sealed off bag.
Do some rooting around and find where the ants could be entering. If you find the entry/ies, use a liquid glue to seal up the hole.
You may not like sprays, but they work the best, and you only spray along base boards or problem areas.
I have heard another alternative is to, this is if you want to go all green about it, place dry rice out. Yes, dry rice. The ants see and detect the rice as being edible. Problem is they don't have enough saliva to turn it into edible food, and they don't know that. In turn, they gather all this food and essentially starve themselves to death. However, this would mean that you are leaving dry rice laying out, and your cat could possibly eat it as well.
Calling or talking to a professional might be a viable option, i believe there is a store in Seattle that just sell home stuff for stopping pests.