The Hamiltonian path problem is a well-known problem in computer science and graph theory. It involves finding a path in a graph that visits each vertex exactly once. Dynamic programming is a technique that can be used to solve this problem efficiently.

The time complexity of solving the Hamiltonian path problem using dynamic programming depends on the number of vertices in the graph. Let's analyze the given options to understand which one is the correct answer.

a) O(N): This option suggests that the problem can be solved in linear time, where N is the number of vertices in the graph. However, finding a Hamiltonian path is known to be an NP-complete problem, which means there is no known polynomial-time algorithm for it. Therefore, option A is incorrect.

b) O(N log N): This option suggests that the problem can be solved in N log N time. However, this is not possible because the problem is NP-complete. Therefore, option B is incorrect.

c) O(N^2): This option suggests that the problem can be solved in quadratic time, where N is the number of vertices in the graph. This is a possibility because dynamic programming can be used to solve the Hamiltonian path problem in O(N^2 2^N) time. However, we need to analyze the last option to make sure it is not a better choice. Therefore, option C is a possibility but may not be the correct answer.

d) O(N^2 2^N): This option suggests that the problem can be solved in O(N^2 2^N) time, where N is the number of vertices in the graph. This is the correct answer because dynamic programming can be used to solve the Hamiltonian path problem in this time complexity. The 2^N factor is due to the exponential number of subsets of vertices that need to be considered. By using dynamic programming, we can avoid redundant computations and solve the problem efficiently. Therefore, option D is the correct answer.

To summarize, the Hamiltonian path problem can be solved using dynamic programming in O(N^2 2^N) time complexity, where N is the number of vertices in the graph. This is the correct answer to the given question.