Every heuristic for the traveling salesman problem that I know of (Nearest-Neighbour, Christofides, Held-Karp, ...) assumes that the triangle inequality holds.
Are there heuristics to solve the tsp that don't assume the triangle inequality and still perform well*?
If there aren't, why is that?
*i.e. don't run for a year for 100 cities and get relatively close to the optimum