We introduce a hierarchy of problems between the Dominating Set problem and the Power Dominating Set (PDS) problem called the ℓ-round power dominating set (ℓ-round PDS, for short) problem. For ℓ = 1, this is the Dominating Set problem, and for ℓ ≥ 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 Dominating Set problem, and that rule (2) is a type of propagation rule that applies iteratively. The ℓ-round PDS problem has the same set of rules as PDS, except we apply rule (2) in "parallel" in at most ℓ − 1 rounds. We prove that ℓ-round PDS cannot be approximated better than 2 log 1−ǫ n even for ℓ = 4 in general graphs. We provide a dynamic programming algorithm to solve ℓ-round PDS optimally in polynomial time on graphs of bounded tree-width. We present a PTAS (polynomial time approximation scheme) for ℓ-round PDS on planar graphs for ℓ = O( log n log log n ). Finally, we give integer programming formulations for ℓ-round PDS.
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−ǫ n in contrast to the logarithmic hardness for the dominating set problem. We give an O( √ 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−ǫ n for directed 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.