Approximability of the upper chromatic number of hypergraphs

Abstract

A C-coloring of a hypergraph H=(X,E) is a vertex coloring φ:X→N such that each edge E∈E has at least two vertices with a common color. The related parameter χ¯(H), called the upper chromatic number of H, is the maximum number of colors in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E∈E induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph.We establish guaranteed polynomial-time approximation ratios for the difference n−χ¯(H), which is 2+2ln(2m) on hypergraphs in general, and 1+lnm on hypertrees. The latter ratio is essentially tight as we show that n−χ¯(H) cannot be approximated within (1−ϵ)lnm on hypertrees (unless NP⊆DTIME(nO(loglogn))). Furthermore, χ¯(H) does not have O(n1−ϵ)-approximation and cannot be approximated within additive error o(n) on the class of hypertrees (unless P=NP).