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 Sthat power dominates all the nodes, where a node vis power dominated if (1) vis in Sor vhas a neighbor in S, or (2) vhas a neighbor wsuch that wand all of its neighbors except vare 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. We use nto denote the number of nodes. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}{n}}$ in contrast to the logarithmic hardness for dominating set. This is the first result separating these two problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. We introduce an extension of PDS called i¾?-round PDS; for i¾?= 1 this is the dominating set problem, and for i¾? i¾? ni¾? 1 this is the PDS problem. Our hardness threshold for PDS also holds for i¾?-round PDS for all i¾? i¾? 4. We give a PTAS for the i¾?-round PDS problem on planar graphs, for $\ell=O(\frac{\log{n}}{\log{\log{n}}})$. We study variants of the greedy algorithm, which is known to work well on covering problems, and show that the approximation guarantees can be i¾?(n), even on planar graphs. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of $2^{\log^{1-\epsilon}{n}}$ for directed acyclic graphs.