Finding large independent sets in polynomial expected time

We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let G(n,p) be a random graph, and let S be a set of k vertices, chosen uniformly at random. Then, let G(0) be the graph obtained by deleting all edges connecting two vertices in S. Finally, an adversary may add edges to G(0) that do not connect two vertices in S, thereby producing the instance G = G(n,p,k)(*) We present an algorithm that on input G = G(n,p,k)(*) finds an independent set of size >= k within polynomial expected time, provided that k >= C(n/p)(1/2) for a certain constant C > 0. Moreover, we prove that in the case k <= (1-epsilon)ln(n)/p this problem is hard.