Exact Algorithms for No-Rainbow Coloring and Phylogenetic Decisiveness

The input to the no-rainbow hypergraph coloring problem is a hypergraph $H$ where every hyperedge has $r$ nodes. The question is whether there exists an $r$-coloring of the nodes of $H$ such that all $r$ colors are used and there is no rainbow hyperedge – i.e., no hyperedge uses all $r$ colors. The no-rainbow hypergraph $r$-coloring problem is known to be NP-complete for $r\ge 3$. The special case of $r=4$ is the complement of the phylogenetic decisiveness problem. Here we present a deterministic algorithm that solves the no-rainbow $r$-coloring problem in $O^{\ast }((r-1{)}^{(r-1)n\mathrm{/}r})$ time and a randomized algorithm that solves the problem in $O^{\ast }((\frac{r}{2}{)}^n)$ time.