Approximation Algorithms and Hardness for Domination with Propagation

• Authors: Ashkan Aazami;Michael D. Stilp
• Venue: APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
• Year: 2007

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 茂戮驴-round PDS; for 茂戮驴= 1 this is the dominating set problem, and for 茂戮驴 茂戮驴 n茂戮驴 1 this is the PDS problem. Our hardness threshold for PDS also holds for 茂戮驴-round PDS for all 茂戮驴 茂戮驴 4. We give a PTAS for the 茂戮驴-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 茂戮驴(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.