  We introduce a hierarchy of problems between the \textsc{Dominating Set}
problem and the \textsc{Power Dominating Set} (PDS) problem called the
$\ell$-round power dominating set ($\ell$-round PDS, for short) problem. For
$\ell=1$, this is the \textsc{Dominating Set} problem, and for $\ell\geq n-1$,
this is the PDS problem; here $n$ denotes the number of nodes in the input
graph. In PDS the goal is to find a minimum size set of nodes $S$ that power
dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in
$S$ or it has a neighbor in $S$, or (2) $v$ has a neighbor $u$ such that $u$
and all of its neighbors except $v$ are power dominated. Note that rule (1) is
the same as for the \textsc{Dominating Set} problem, and that rule (2) is a
type of propagation rule that applies iteratively. The $\ell$-round PDS problem
has the same set of rules as PDS, except we apply rule (2) in ``parallel'' in
at most $\ell-1$ rounds. We prove that $\ell$-round PDS cannot be approximated
better than $2^{\log^{1-\epsilon}{n}}$ even for $\ell=4$ in general graphs. We
provide a dynamic programming algorithm to solve $\ell$-round PDS optimally in
polynomial time on graphs of bounded tree-width. We present a PTAS (polynomial
time approximation scheme) for $\ell$-round PDS on planar graphs for
$\ell=O(\tfrac{\log{n}}{\log{\log{n}}})$. Finally, we give integer programming
formulations for $\ell$-round PDS.
