Coloring Sparse Random k-Colorable Graphs in Polynomial Expected Time

Abstract

Feige and Kilian [5] showed that finding reasonable approximative solutions to the coloring problem on graphs is hard. This motivates the quest for algorithms that either solve the problem in most but not all cases, but are of polynomial time complexity, or that give a correct solution on all input graphs while guaranteeing a polynomial running time on average only. An algorithm of the first kind was suggested by Alon and Kahale in [1] for the following type of random k-colorable graphs: Construct a graph $\mathcal{G}_{n,p,k}$ on vertex set V of cardinality n by first partitioning V into k equally sized sets and then adding each edge between these sets with probability p independently from each other. Alon and Kahale showed that graphs from $\mathcal{G}_{n,p,k}$ can be k-colored in polynomial time with high probability as long as p ≥ c/n for some sufficiently large constant c. In this paper, we construct an algorithm with polynomial expected running time for k = 3 on the same type of graphs and for the same range of p. To obtain this result we modify the ideas developed by Alon and Kahale and combine them with techniques from semidefinite programming. The calculations carry over to general k.