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An exact semidefinite programming approach for the max-mean dispersion problem

Author

Listed:
  • Michele Garraffa

    (DAUIN, Politecnico di Torino)

  • Federico Della Croce

    (DAUIN, Politecnico di Torino)

  • Fabio Salassa

    (DAUIN, Politecnico di Torino)

Abstract

This paper proposes an exact algorithm for the Max-Mean dispersion problem ( $$Max-Mean DP$$ M a x - M e a n D P ), an NP-Hard combinatorial optimization problem whose aim is to select the subset of a set such that the average distance between elements is maximized. The problem admits a natural non-convex quadratic fractional formulation from which a semidefinite programming (SDP) relaxation can be derived. This relaxation can be tightened by means of a cutting plane algorithm which iteratively adds the most violated triangular inequalities. The proposed approach embeds the SDP relaxation and the cutting plane algorithm into a branch and bound framework to solve $$Max-Mean DP$$ M a x - M e a n D P instances to optimality. Computational experiments show that the proposed method is able to solve to optimality in reasonable time instances with up to 100 elements, outperforming other alternative approaches.

Suggested Citation

  • Michele Garraffa & Federico Della Croce & Fabio Salassa, 2017. "An exact semidefinite programming approach for the max-mean dispersion problem," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 71-93, July.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:1:d:10.1007_s10878-016-0065-1
    DOI: 10.1007/s10878-016-0065-1
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    References listed on IDEAS

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    1. Prokopyev, Oleg A. & Kong, Nan & Martinez-Torres, Dayna L., 2009. "The equitable dispersion problem," European Journal of Operational Research, Elsevier, vol. 197(1), pages 59-67, August.
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    4. Martí, Rafael & Gallego, Micael & Duarte, Abraham, 2010. "A branch and bound algorithm for the maximum diversity problem," European Journal of Operational Research, Elsevier, vol. 200(1), pages 36-44, January.
    5. C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
    6. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
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    Cited by:

    1. Martí, Rafael & Martínez-Gavara, Anna & Pérez-Peló, Sergio & Sánchez-Oro, Jesús, 2022. "A review on discrete diversity and dispersion maximization from an OR perspective," European Journal of Operational Research, Elsevier, vol. 299(3), pages 795-813.
    2. Spiers, Sandy & Bui, Hoa T. & Loxton, Ryan, 2023. "An exact cutting plane method for the Euclidean max-sum diversity problem," European Journal of Operational Research, Elsevier, vol. 311(2), pages 444-454.

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