Heuristics for Finding Large Independent Sets, with Applications to Coloring Semi-Random Graphs

We study a semi-random graph model for finding independent sets. For > 0, an n-vertex graph with an independent set S of size n is constructed by blending random and adversarial decisions. Randomly and independently with probability p, each pair of vertices, such that one is in S and the other is not, is connected by an edge. An adversary can then add edges arbitrarily (provided that S remains an independent set). The smaller p is, the larger the control the adversary has over the semi-random graph. We design heuristics that with high probability recover S when p > (1 + ) lnn=jSj, for any constant > 0. We show that when p < (1 ) lnn=jSj, an independent set of size jSj cannot be recovered, unless NP BPP . We use our results to obtain greatly improved coloring algorithms for the model of k-colorable semi-random graphs introduced by Blum and Spencer.