Algorithms for coloring semi-random graphs

Abstract

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.