Approximation Algorithms and Hardness for Domination with Propagation
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
Generalized power domination of graphs
Discrete Applied Mathematics
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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 $2^{\log^{1-\epsilon}n}$ 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 $2^{\log^{1-\epsilon}n}$ 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.