Suppose that you have a magic box that can solve the traveling salesman problem - you provide the box with a cost matrix where every non-diagonal entry is a positive integer, and it gives you the minimum cost of a tour of the vertices. (The box does not say what the tour looks like.) Explain how you be able to use this magic box to determine whether an arbitrary graph is Hamiltonian. How could you use it to determine whether a graph has a Hamiltonian path?