Finding shortest circuit in a graph that visits X nodes at least once

Question

Even though I\'m still a beginner, I love solving graph related problems (shortest path, searches, etc). Recently I faced a problem like this :

Given

Answer 1

Here is a solution which likely to be fast enough:
1)Run shortest path search algorithm from every blue node(this can be done in O(X * (E log N))) to compute pairwise distances.
2)Build a new graph with zero vertex and blue vertices only(X + 1 vertices). Add edges using pairwise distances computed during the first step.
3)The new graph is small enough to use dynamic programming solution for TSP(it has O(X^2 * 2^X) time complexity).