Improved algorithms for coloring random graphs

Abstract

Consider the problem of k-coloring random k-colorable graphs. The random graphs are drawn from the G(n,p,k) model. In this model, an adversary splits the vertices into k color classes each of size Ω(n), and then for each pair u,v of vertices belonging to different color classes, he includes the edge {u,v} with probability p. We give algorithms for coloring random graphs from G(n,p,k) with p ≥ n−1+e where e > 1/3 is any positive constant. The failure probability of our algorithms is exponentially low, i.e. \(e^{ - n^\delta } \)for some positive constant δ. This improves the previously known algorithms which have only polynomially low failure probability. We also show how the algorithms given by Blum and Spencer can be modified to have exponentially low failure probability provided some restrictions on edge probabilities are introduced. We introduce a technique for converting almost succeeding algorithms into algorithms with polynomial expected running time and which surely k-color the input random graph. Using these two results, we derive polynomial expected time algorithms for k-coloring random graphs from G(n,p,k) with p ≥ n−1+e where e is any constant greater than 0.4. Our results improve the previously known results.