Approximating the Achromatic Number Problem on Bipartite Graphs

Abstract

The achromatic number of a graph is the largest number of colors needed to legally color the vertices of the graph so that adjacent vertices get different colors and for every pair of distinct colors c1,c2 there exists at least one edge whose endpoints are colored by c1,c 2. We give a greedy O(n4/5) ratio approximation for the problem of finding the achromatic number of a bipartite graph with n vertices. The previous best known ratio was n · log log n/log n [12]. We also establish the first non-constant hardness of approximation ratio for the achromatic number problem; in particular, this hardness result also gives the first such result for bipartite graphs. We show that unless NP has a randomized quasi-polynomial algorithm, it is not possible to approximate achromatic number on bipartite graph within a factor of (ln n)1/4-ε. The methods used for proving the hardness result build upon the combination of one-round, two-provers techniques and zero-knowledge techniques inspired by Feige et.al. [6]. © Springer-Verlag 2003.