Minimum Coloring k-Colorable Graphs in Polynomial Average Time

Abstract

We present algorithms for minimum (χ(G)-) coloring k-colorable graphs drawn from random and semi-random models. In both models, an adversary initially splits the vertices into k color classes V1,…,Vk of sizes Ω(n) each. In the first model, each potential edge joining vertices in different classes is included with independent probability p. In the latter model, the adversary fixes an ordering of all potential edges and then for each i, the ith edge (according to this ordering) is included with probability p(ei,S)≥p,S⊆{1,…,i−1}. Thus, the probability of inclusion for the ith edge depends on the outcomes on previous i−1 edges. Semi-random models were introduced as a way of striking a balance between random graphs and worst-case adversaries. Our algorithms run in polynomial time on average. χ(G)-coloring is harder than k-coloring because even a “short certificate” is not presently known for the optimality of a coloring. Our algorithms work as long as p≥n−α(k)+ϵ for the semi-random model and p≥n−γ(k)+ϵ for the random model where α(k)=(2k)/((k−1)(k+2)), γ(k)=(2k)/(k2−k+2), and ϵ is any positive constant.