SDP relaxation of independent set
I believe SDP is the only known technique to solve the maximum independent set problem on perfect graphs. To get the independent set, one could do the following. Guess if a vertex is in the independent set, delete it and solve the SDP. If it returns the same value, then there is an independent set without this vertex. So, make this vertex adjacent to all other vertices, and continue. This should give you an actual independent set. 

Otherwise, we have identified one vertex of the independent set, and we can remove it and continue on the remaining graph.