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Complexity of determining the most vital elements for the p-median and p-center location problems

Author

Listed:
  • Cristina Bazgan

    (Université Paris-Dauphine
    Institut Universitaire de France)

  • Sonia Toubaline

    (Université Paris-Dauphine)

  • Daniel Vanderpooten

    (Université Paris-Dauphine)

Abstract

We consider the k most vital edges (nodes) and min edge (node) blocker versions of the p-median and p-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) p-median (respectively p-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the p-median (respectively p-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker p-median (respectively p-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the p-median (respectively p-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges p-median and k most vital edges p-center are NP-hard to approximate within a factor $\frac{7}{5}-\epsilon$ and $\frac{4}{3}-\epsilon$ respectively, for any ϵ>0, while k most vital nodes p-median and k most vital nodes p-center are NP-hard to approximate within a factor $\frac{3}{2}-\epsilon$ , for any ϵ>0. We also show that the complementary versions of these four problems are NP-hard to approximate within a factor 1.36.

Suggested Citation

  • Cristina Bazgan & Sonia Toubaline & Daniel Vanderpooten, 2013. "Complexity of determining the most vital elements for the p-median and p-center location problems," Journal of Combinatorial Optimization, Springer, vol. 25(2), pages 191-207, February.
    • Handle: RePEc:spr:jcomop:v:25:y:2013:i:2:d:10.1007_s10878-012-9469-8
    DOI: 10.1007/s10878-012-9469-8
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    Cited by:

    1. Stephen R. Chestnut & Rico Zenklusen, 2017. "Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 144-166, January.
    2. Qiao Zhang & Xiucui Guan & Panos M. Pardalos, 2021. "Maximum shortest path interdiction problem by upgrading edges on trees under weighted $$l_1$$ l 1 norm," Journal of Global Optimization, Springer, vol. 79(4), pages 959-987, April.
    3. T. N. Dinh & M. T. Thai & H. T. Nguyen, 2014. "Bound and exact methods for assessing link vulnerability in complex networks," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 3-24, July.
    4. Nicolas Fröhlich & Stefan Ruzika, 2022. "Interdicting facilities in tree networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 95-118, April.

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