  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 $r \geq 3$. 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 $O^*((r-1)^{(r-1)n/r})$ time and a randomized algorithm
that solves the problem in $O^*((\frac{r}{2})^n)$ time.
