Approximability of the upper chromatic number of hypergraphs

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 can be used 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 + 2 ln(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− ǫ) lnm on hypertrees (unless NP⊆DTIME(nO(log log n))). Furthermore, χ(H) does not have O(n1−ǫ)-approximation and cannot be approximated within additive error o(n) on the class of hypertrees (unless P = NP).