Approximation algorithms and hardness for domination with propagation

The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-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 $v$ has a neighbor in $S$, or (2) $v$ has a neighbor $w$ such that $w$ and all of its neighbors except $v$ are power dominated. We show a hardness of approximation threshold of $2^{{\log }^{1-ϵ}n}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of $2^{{\log }^{1-ϵ}n}$ for directed \emph{acyclic} graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.