Look in the yellow pages or similar for "mudjacking". The process consists of a contractor drilling some 1 1/2" holes in the slab and injecting a cement/sand/mud mixture under the slab to level. After leveling, the holes are patched. It is a old technique that is well suited to level and/or correct drainage of slabs, sidwalks. Usually, it is a fairly quick two man job
It is frequently listed under concrete contractors as slab raising, concrete raising, slab leveling, mudjacking, etc.
It is surprising what results you can get. Certainly much cheaper, faster and less mess than a tear-out and replacement.