Cats have to do their business someplace. Cut a hole in a rubbermaid container with high sides and set it in half a cardboard box. The hole should face the interior of the cardboard box.
This arrangement is cheap and works far better than anything I have ever seen. It works well for even giant size cats if you get a big enough container with high sides. The cat is forced to walk on the cardboard which tends to get rid of any litter before he gets on your floor. Accidents are almost totally eliminated. I am sure the idea is not original to me because it just works too good and is too obvious. Expensive cat litter boxes are just not worth the money.