I wasn't aware that the "NP-heuristics" are able to guaranty such good results like Christofides does.
I only knew because I took a course from a guy (EDIT: Bill Cook
, who literally wrote the book on the TSP) whose research specialty was the TSP. Another cool thing about it is that you can find good lower bounds on the length (based on the minimum spanning tree length), so you can use those plus a heuristic (i.e. upper bound) to iteratively find the true solution.
I have no idea whether "good-enough" solutions to the TSP can be translated to "good-enough" solutions to other NP-complete problems, but that's an interesting thought. Many of the ways of reducing one NP-complete problem to another are fairly bizarre, so solution quality might not translate.