> TSP is actually very amenable to heuristics and state of the art branch-and-bound algorithms can often find optimal solutions even for instances with thousand

> TSP is actually very amenable to heuristics and state of the art branch-and-bound algorithms can often find optimal solutions even for instances with thousands of points. In own research (inverse problems related to signal recovery), I've noticed that a lot of NP-hard problems are actually easily solvable given that the data has some kind of structure to it. I know very little about computational complexity theory, but I've read some papers describing how there is often a phase transition where a high SNR puts the problem in P territory, but a sufficiently low SNR moves the problem moves to NP territory (but it's still solvable). Beyond the phase transition, a solution is information theoretically impossible. I wonder if most of these real world TSP problems actually lie in P, but I don't know how one would go about showing that. (I haven't performed a literature search, but I wouldn't be surprised if someone has already demonstrated this.)