  Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G)
\rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with
$f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number
$\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs.
