This problem came out of my recent [blog post](http://blog.computationalcomplexity.org/2014/01/is-traveling-salesman-np-complete.html), suppose you are given a TSP tour, is it co-NP-complete to determine if it is a minimal one?

More precisely is the following problem NP-complete:

Instance: Given a complete graph G with edges weighted with positive integers and a simple cycle C that visits all the nodes of G.

Question: Is there a simple cycle D that visits all the nodes of G such that the total weight of all the edges of D in G is strictly less than the total weight of all the edges of C in G?