Domination in graphs with bounded propagation: algorithms, formulations and hardness results

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.