systems are almost as deep and wide as a tree. I don't think one offending root would be too big of a deal.

I think I'd dig down on both sides of my walk until I found the offending root. Then I'd probably sawzall it or something similar. If you cut all the way through I'd probably just leave the root there and let it slowly compose. You could "dig" it out with a water tube/hose approach and then backfill with a sand or something similar. But I'd only do this if the cement really needs to go back down in place.

Just my two cents. I'm sure there are a lot more with more wisdom on this board.