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Improved Algorithms and Complexity Results for Power Domination in Graphs

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Abstract

The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum-size set P⊆V such that all vertices in V are “observed” by the vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” For treewidth being upper-bounded by a constant, we achieve a linear-time algorithm. In particular, we present a simplified linear-time algorithm for Power Dominating Set in trees. Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and it cannot be better approximated than Dominating Set.

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Authors and Affiliations

  1. Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Jiong Guo & Rolf Niedermeier

  2. Abteilung Informatik/Wirtschaftsinformatik, Fachbereich IV, Universität Trier, 54286, Trier, Germany

    Daniel Raible

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  1. Jiong Guo
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  2. Rolf Niedermeier
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  3. Daniel Raible
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Correspondence to Jiong Guo.

Additional information

A preliminary version of this paper appears in the proceedings of the 15th International Symposium on Fundamentals of Computation Theory (FCT’05), vol. 3623 in LNCS, pp. 172–184, Springer (2005). Work supported by the Deutsche Forschungsgemeinschaft (DFG), Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4. Significant portions of this work were done while all authors were affiliated with the Universität Tübingen.

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Guo, J., Niedermeier, R. & Raible, D. Improved Algorithms and Complexity Results for Power Domination in Graphs. Algorithmica 52, 177–202 (2008). https://doi.org/10.1007/s00453-007-9147-x

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