Only place the clean parts into the recycling cart.

The parts stained with grease, cheese, sauce or other pizza toppings belong in the trash.

In other words if the top is clean and the bottom is greasy, rip the top off and recycle that and toss the greasy part into the trash.

Recycling rules vary around the country. The fact that recycling rules are regional is not something widely appreciated.

For the paper mills that accept the cardboard material collected from Madison carts, the greasy, food-stained parts are still considered a contaminant.

can i compost the greasy parts of the pizza box?

Not in your backyard bins.  Clean cardboard is something that you could use as a carbon source for composting (though you should tear it up into pieces).  The greasy, cheesy, toppings-stained parts could attract unwanted critters like rats to your compost bin or contribute to foul odors.