  The $k$-rainbow independent domination number of a graph $G$, denoted
$\gamma_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two
vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in
$V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This
domination invariant was proposed by {\v{S}}umenjak, Rall and Tepeh in (Applied
Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the
problem of computing the independent domination number of the generalized prism
$G {\Box} K_k$ to an integer labeling problem on $G$. They proved a
Nordhaus-Gaddum-type theorem: $5\leq \gamma_{\rm rik}(G)+\gamma_{\rm
rik}(\overline{G})\leq n+3$ for every graph $G$ of order $n\geq 3$, where
$\overline{G}$ is the complement of $G$. In this paper, we improve this result
by showing that if $G$ is not isomorphic to the 5-cycle, then $5\leq
\gamma_{\rm rik}(G)+\gamma_{\rm rik}(\overline{G})\leq n+2$. Moreover, we show
that the problem of deciding whether a graph has a $k$-rainbow independent
dominating function of a given weight is $\mathcal{NP}$-complete. Our results
respond some open questions proposed by \v{S}umenjak, et al.
