Is there an algorithm to compute the shortest Hamiltonian path in an directed graph from one point to another in polynomial time? I am wondering whether the Shortest Hamiltonian Path (SHP) problem is NP-Complete, because I couldn't find a way of solving it in a polynominal time. if it is NP_Complete, it will be so kind of you to give me the proof or the relevent papers,thanks a lot 

the problem explanation: Given a directed,weighted graph with n vertices, find the shortest hamiltonian path with end vertices v and u.
by the way,the graph must exit a hamiltonian path from v to u.
Here's a small hint.

Let $G$ be a directed graph with unit weights, and $u$ and $v$ be two vertices.  Suppose that I could solve SHP in polynomial time, say, $O(n^c)$.

Now, say that I would like to solve Hamiltonian Path on a directed graph.  Let me spend $O(n^2)$ time to "guess" your vertices $u$ and $v$, and for each of them run your algorithm for SHP.  Then, I would in $O(n^{c+2})$ time have either found a Hamiltonian path in $G$, or conclude that $G$ is not Hamiltonian.

That would mean that P = NP.

Now, this doesn't prove that your problem in NP-complete.  For that, you need to come up with a reduction.  But it does tell you that you should probably not spend too much time looking for a polynomial time algorithm.