Minimum Coloring Random and Semi-Random Graphs in Polynomial Expected Time

We present new algorithms for k-coloring and minimum (x(G)-) coloring random and semi-random kcolorable graphs in polynomial expected time. The random graphs are drawn from the G(n,p, k) model and the semi-random graphs are drawn from the G s ~ ( n , p , k) model. In both models, an adversary initially splits the n vertices into IC color classes, each of size @(n). Then the edges between vertices in different color classes are chosen one by one, according to some probability distribution. The model G s ~ ( n , p , k ) was introduced by Blum [3] and with respect to randomness, it lies between the random model G(n,p, k) where all edges are chosen with equal probability and the worst-case model. This extended abstract consists of two parts. In Part I, we propose a general methodology for designing algorithms for k-coloring random graphs from G(n,p, k). Using this, we derive new algorithms for k-coloring G E G(n,p, k) for p 2 n-’+‘ where E is any constant greater than 1/4. Our algorithms run in polynomial time on the average. This improves the results of [13] where E was required to be above 0.4. In Part 11, we present polynomial average time algorithms for minimum coloring semi-random graphs from G s ~ ( n , p , k) for p 2 n-a(k)+e, where a(k) = (2IC)/((k l ) ( k + 2)) and E is any positive constant. We also present polynomial average time algorithms for X(G)-coloring random graphs from G(n,p, k) , with p 2 n--f(k)+a where y(k) = (2k)/(k2 k + 2). The problem of X(G)-coloring is harder than the k-coloring problem because every k-colorable graph has a ”short certificate” for k-colorability, but there are many kcolorable graphs with x(G) = k for which there is no known short certificate for the fact x(G) = k.