Questions:
Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation of the edge weights of an euclidean TSP?
If sufficiently small perturbations of the edge weights do not harm a PTAS, the PTAS can't be tied to euclidicity of the instance and there should be more general conditions for metric TSPs that allow a PTAS; are such conditions known or being investigated?