Approximability of the upper chromatic number of hypergraphs

A C-coloring of a hypergraph ${\cal H}=(X,{\cal E})$ is a vertex coloring $φ: X\to {\mathbb{N}}$ such that each edge $E\in{\cal E}$ has at least two vertices with a common color. The related parameter $\overlineχ({\cal H})$, called the upper chromatic number of ${\cal H}$, is the maximum number of colors can be used in a C-coloring of ${\cal H}$. A hypertree is a hypergraph which has a host tree $T$ such that each edge $E\in {\cal E}$ 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 $n-\overlineχ({\cal H})$, which is $2+2 \ln (2m)$ on hypergraphs in general, and $1+ \ln m$ on hypertrees. The latter ratio is essentially tight as we show that $n-\overlineχ({\cal H})$ cannot be approximated within $(1-ε) \ln m$ on hypertrees (unless ${\sf NP} \subseteq {\sf DTIME} (n^{{\cal O}(log\;log\; n)})$). Furthermore, $\overlineχ({\cal H})$ does not have ${\cal O}(n^{1-ε})$-approximation and cannot be approximated within additive error $o(n)$ on the class of hypertrees (unless ${\sf P}={\sf NP}$).

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