SDP relaxation of independent set I'm looking at page 28 of Lovasz "Semidefinite programs and combinatorial optimization" and it gives the following approximation of independence number of the graph

$$\max u&#39; Z u$$
subject to
$$Z\succ 0$$
$$Z_{ij}=0 \ \forall ij\in E(G)$$
$$tr(Z)=1$$

Can I get independent set (or something close to an independent set) directly from the solution of SDP relaxation? Lovasz says that SDP is the only known way to solve this problem exactly for perfect graphs, is that still true?

Clarification: there's a similar SDP relaxation for the size of maximum cut, and I can get the full solution (the actual cut, rather than its size) by taking square root of Z and doing randomized rounding (Ch.6 of Williamson/Shmoys book). I'm wondering if there's a similar technique for this problem
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.