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A fast approximation algorithm for the maximum 2-packing set problem on planar graphs

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Abstract

Given an undirected graph \(G=(V, E)\), a subset \(S\subseteq V\) is a 2-packing set if, for any pair of vertices \(u,v \in S\), the shortest path between them is at least three-edge long. Finding a 2-packing set of maximum cardinality is an NP-hard problem for arbitrary graphs. This paper proposes an approximation algorithm for the maximum 2-packing set problem for planar graphs. We show that our algorithm is at least \(\frac{\lambda -2}{\lambda }\) of the optimal (i.e. the approximation ratio is \(\frac{\lambda }{\lambda -2}\)), where \(\lambda\) is a constant related to how the proposed algorithm decomposes the input graph into smaller subgraphs. Then, we improve the solution given by our approximation algorithm by adding some vertices to the solution. Experimentally, we show that our improved algorithm computes a near-optimal 2-packing set. This algorithm is the first approximation algorithm for the maximum 2-packing set to the best of our knowledge.

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Data Availability

Graph instances are available at the following link:https://github.com/trejoel/Approximate2Packing

Code Availability

Source code is available at the following link: https://github.com/trejoel/Approximate2Packing

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Acknowledgements

This work was partially supported by the National Council of Science and Technology, Mexico (CONACYT) with the Frontier Science Project no. 304006.

Funding

The National Council of Science and Technology, Mexico (CONACYT), through the Frontier Science Project no. 304006. A. Flores-Lamas is supported by the Engineering and Physical Sciences Research Council, EPSRC [EP/T00021X/1].

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Authors and Affiliations

  1. Department of Computer Sciences, CONACYT-Center for Research in Mathematics, Mérida, Mexico

    Joel Antonio Trejo-Sánchez

  2. Department of Computer Sciences, Universidad Autónoma de Yucatán, Mérida, Mexico

    Francisco A. Madera-Ramírez & José Luis López-Martínez

  3. Department of Computer Sciences, Center for Scientific Research and Higher Education of Ensenada (CICESE), Ensenada, Mexico

    José Alberto Fernández-Zepeda

  4. Department of Computer Sciences, Royal Holloway University of London, Egham Hill, United Kingdom

    Alejandro Flores-Lamas

Authors
  1. Joel Antonio Trejo-Sánchez
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  2. Francisco A. Madera-Ramírez
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  3. José Alberto Fernández-Zepeda
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  4. José Luis López-Martínez
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  5. Alejandro Flores-Lamas
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Contributions

All the authors contributed equally to this work —Joel Antonio Trejo-Sánchez: algorithm’s conception, design, and correctness proof. Francisco Alejandro Madera-Ramirez: algorithm’s conception, design and implementation. Jose Luis Lopez-Martinez: algorithm’s implementation and experiments’ design. José Alberto Fernandez-Zepeda: algorithm’s design and correctness proof. Alejandro Flores-Lamas: algorithm’s correctness proof and experiments’ design.

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Correspondence to Joel Antonio Trejo-Sánchez.

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Trejo-Sánchez, J.A., Madera-Ramírez, F.A., Fernández-Zepeda, J.A. et al. A fast approximation algorithm for the maximum 2-packing set problem on planar graphs. Optim Lett 17, 1435–1454 (2023). https://doi.org/10.1007/s11590-022-01939-w

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