Are there NP-complete problems with polynomial expected time solutions?
There is a polynomial time algorithm for finding Hamiltonian cycles on random graphs, that succeeds asymptotically with the same probability that a Hamiltonian cycle exists. Of course, this problem is NP-hard in the worst case. 

They also show that a dynamic programming algorithm which is always guaranteed to find a Hamiltonian cycle, if it exists, has polynomial expected running time, if the input distribution is uniformly random over the set of all $n$ vertex graphs.

Reference: "An algorithm for finding Hamilton cycles in random graphs"

Bollobas, Fenner, Frieze

http://portal.acm.org/citation.cfm?id=22145.22193