Can I or should I prune a problematic root of my honey locust?

I have a honey locust that was planted in a street lawn strip 18 years ago in Colorado. It has a root that is growing under the sidewalk and halfway (at least) up my driveway. The driveway has been heaving for years and I have dug around thinking it was a nearby ash tree causing the problem, but I never found a root causing it. Today I see a honey locust sucker come up in a crack between the driveway slab. Looking from that spot and making a mental line directly to the tree, I see that the heaving all takes place along that path, so it much be this tree causing the problem. I would like to sever the root where it meets the sidewalk, to prevent further under the driveway and lessen the chance I will need to rip it up and have it re-poured. My fear is what it will do to the tree. I feel like it's a damned if you do, damned if you don't situation. Do I let the tree do what trees do, and eventually make the driveway worse, when I will them probably need to cut it anyway? Or do I cut the root to stop further damage and wait and see what effect this will have on the tree? I also wonder if doing nothing will just cause problems for the tree anyway because it's rooting out under both the sidewalk/driveway and the road.