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

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 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( logn log logn ). Finally, we give integer programming formulations for l-round PDS.