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

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  • Ashkan Aazami

    (University of Waterloo)

Abstract

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 $2^{\log^{1-\epsilon}{n}}$ 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 $\ell=O(\frac{\log{n}}{\log{\log{n}}})$ . Finally, we give integer programming formulations for ℓ-round PDS.

Suggested Citation

  • Ashkan Aazami, 2010. "Domination in graphs with bounded propagation: algorithms, formulations and hardness results," Journal of Combinatorial Optimization, Springer, vol. 19(4), pages 429-456, May.
  • Handle: RePEc:spr:jcomop:v:19:y:2010:i:4:d:10.1007_s10878-008-9176-7
    DOI: 10.1007/s10878-008-9176-7
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    Citations

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    Cited by:

    1. Chao Wang & Lei Chen & Changhong Lu, 2016. "$$k$$ k -Power domination in block graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 865-873, February.
    2. Daniela Ferrero & Leslie Hogben & Franklin H. J. Kenter & Michael Young, 2017. "Note on power propagation time and lower bounds for the power domination number," Journal of Combinatorial Optimization, Springer, vol. 34(3), pages 736-741, October.
    3. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 0. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-19.
    4. Chung-Shou Liao, 2016. "Power domination with bounded time constraints," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 725-742, February.
    5. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 2020. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 482-500, August.
    6. Boris Brimkov & Derek Mikesell & Illya V. Hicks, 2021. "Improved Computational Approaches and Heuristics for Zero Forcing," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1384-1399, October.
    7. Eduardo Álvarez-Miranda & Markus Sinnl, 2020. "A branch-and-cut algorithm for the maximum covering cycle problem," Annals of Operations Research, Springer, vol. 284(2), pages 487-499, January.
    8. Boris Brimkov & Derek Mikesell & Logan Smith, 2019. "Connected power domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 292-315, July.
    9. Liao, Chung-Shou & Hsieh, Tsung-Jung & Guo, Xian-Chang & Liu, Jian-Hong & Chu, Chia-Chi, 2015. "Hybrid search for the optimal PMU placement problem on a power grid," European Journal of Operational Research, Elsevier, vol. 243(3), pages 985-994.
    10. Brimkov, Boris & Fast, Caleb C. & Hicks, Illya V., 2019. "Computational approaches for zero forcing and related problems," European Journal of Operational Research, Elsevier, vol. 273(3), pages 889-903.

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