For roots breaking driveway cement, the first thing is to separate them from the tree. Dig down and expose the roots. Then get a saw, such as a bow saw or a power tool like a sawzall (similar to a giant saber saw) and cut through the root. Be careful if using a chain saw because the chain dulls quickly when encountering dirt.
According to David Soulliere (Mr. Landscaper), the roots will take over 10 years to reestablish to the size they are now. What you should be checking for is why did the roots go under the cement? There must be a source of water or nutrients under the cement that are drawing the roots there. This could be indicating a broken sewer line or some other situation under your garage you should be addressing.
Once the root is cut only small feeder roots can start to grow under the cement. You could use some vegetation killer to help prevent further root development in that area. If you do use vegetation killer, be careful because it could kill your tree if too much of it gets to the tree.
You should also use caution when messing with trees on the lot line. Half of that tree is your neighbor’s and killing it could become your liability. You can only cut that part of the tree that comes into your yard. Check with your city forestry department before taking any chances.