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Extended formulations for perfect domination problems and their algorithmic implications

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  • do Forte, Vinicius L.
  • Hanafi, Saïd
  • Lucena, Abilio

Abstract

Given an undirected graph G=(V,E), a subset D⊆V is called a vertex dominating set (VDS) if every vertex of V either belongs to D or is adjacent to a vertex of D. Additionally, a VDS D is called perfect if every vertex of V∖D is adjacent to a single vertex of D. Finally, the Perfect Vertex Domination Problem (PVDP) asks for a perfect VDS D with the smallest cardinality possible. Domination extends very naturally to the edges of G=(V,E) and the Perfect Edge Domination Problem (PEDP) asks for a perfect edge dominating set with as few edges as possible. We propose new formulations for PVDP and PEDP. They rely on structural properties of perfect dominating sets and are computationally compared with their counterparts from the literature. For the new PEDP formulation, in particular, running times for standard state-of-the-art mixed integer programming codes are shown to frequently lead to speed-ups of orders of magnitude, over their corresponding performances for the remaining PEDP formulations.

Suggested Citation

  • do Forte, Vinicius L. & Hanafi, Saïd & Lucena, Abilio, 2023. "Extended formulations for perfect domination problems and their algorithmic implications," European Journal of Operational Research, Elsevier, vol. 310(2), pages 566-581.
    • Handle: RePEc:eee:ejores:v:310:y:2023:i:2:p:566-581
    DOI: 10.1016/j.ejor.2023.03.022
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    References listed on IDEAS

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    1. Austin Buchanan & Je Sang Sung & Sergiy Butenko & Eduardo L. Pasiliao, 2015. "An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 178-188, February.
    2. Bernard Gendron & Abilio Lucena & Alexandre Salles da Cunha & Luidi Simonetti, 2014. "Benders Decomposition, Branch-and-Cut, and Hybrid Algorithms for the Minimum Connected Dominating Set Problem," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 645-657, November.
    3. Gianni Codato & Matteo Fischetti, 2006. "Combinatorial Benders' Cuts for Mixed-Integer Linear Programming," Operations Research, INFORMS, vol. 54(4), pages 756-766, August.
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