Coloring semi-random graphs in polynomial expected time

Abstract

We present algorithms for coloring k-colorable semi-random graphs in polynomial expected time. The semi-random graphs are drawn from the G _SB (n,p,k) model. This model was introduced by Blum [1] and with respect to randomness, this model lies between the random model G(n,p, k) where all edges are chosen with equal probability and the worst-case model. In this model, an adversary splits the n vertices into k color classes, each of size Θ(n). Then, the adversary chooses an ordering of all edges {itu, v} such that u and v belong to different color classes. Based on this ordering, he considers each edge for inclusion by picking a bias p _uv between p and 1 — p of a coin which is flipped to determine whether the edge {itu, v} is placed in the graph. The later choices of the adversary may depend on the previous coin tosses. The probability p is called the noise rate of the source.

We give polynomial expected time algorithms for coloring semi-random graphs from G _SB (n,p,k) for p>-n ^−α+ε, where α = (2k)/((k−1)(k+2)) and ε > 0 is any constant. The semi-random model is a generalization of the random model G(n,p, k) and hence it is more difficult to develop algorithms for coloring semi-random graphs. Ours is the first result of this kind for the semi-random model.