At home we have two cats and a dog... between the three of them I spend a few hours each week cleaning up the dog crap in the yard and cleaning out the cats litter box.
Currently the cats both use the one box. A typical box from the pet store it is fully enclosed with a flap door.
So I read once that cats are toilet trainable, so here's the plan.
I have cut out the base of an old mop bucket I had lying around the house.
Step 1: get the cat comfortable going in the bucket
Step 2: move bucket close to toilet
Step 3: Place the bucket inside the toilet bowl with litter
Step 4: Slowly reduce the litter amount
Step 5: When the cat is used to no litter at all I think problem is almost solved
If I get to this point i'll be happy.