I don't understand how travenlling salesman is NP-complete. In particular, how can you possibly verify that a given route is the shortest one in polynomial time?
The TSP isn't NP-complete - it's NP-hard. But the question: Is there a path shorter than x is NP-complete. A problem is NP-hard iff it can be used to solve an NP-complete problem in O(n) time.