Algorithms for Coloring Semi-random Graphs

Polynomial average time algorithms for $k$-coloring semi-random $k$-colorable graphs are presented and analyzed. Semi-random graphs are a generalization of random graphs and in terms of randomness, this model lies between random graphs and worst-case model

Algorithms for Coloring Semi-random Graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require $n^\delta (\delta>0)$ colors even for bounded chromatic k-colorable for fixed k n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms w always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19, 204-234 1995] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work p \ge n^{-{\alpha (k)}+\epsilon}} Where $\alpha(k) = \frac{(2k)}{((k-1)(k+2))}$ and \epsilon is a positive constant

Coloring semi-random graphs in polynomial expected time

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 A. Blum (1990) 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 {u,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 {u,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^{-\alpha+&epsiv};, 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 mode