A simple and effective algorithm for the MaxMin diversity problem

  • Daniel Cosmin Porumbel
  • Jin-Kao Hao
  • Fred Glover


The challenge of maximizing the diversity of a collection of points arises in a variety of settings, including the setting of search methods for hard optimization problems. One version of this problem, called the Maximum Diversity Problem (MDP), produces a quadratic binary optimization problem subject to a cardinality constraint, and has been the subject of numerous studies. This study is focused on the Maximum Minimum Diversity Problem (MMDP) but we also introduce a new formulation using MDP as a secondary objective. We propose a fast local search based on separate add and drop operations and on simple tabu mechanisms. Compared to previous local search approaches, the complexity of searching for the best move at each iteration is reduced from quadratic to linear; only certain streamlining calculations might (rarely) require quadratic time per iteration. Furthermore, the strong tabu rules of the drop strategy ensure a powerful diversification capacity. Despite its simplicity, the approach proves superior to most of the more advanced methods from the literature, yielding optimally-proved solutions for many problems in a matter of seconds and even attaining a new lower bound.


Maximum diversity MaxMin diversity Tabu search 


  1. Andrade, M., Andrade, P., Martins, S., & Plastino, A. (2005). Lecture notes in computer science: Vol. 3503. GRASP with path-relinking for the maximum diversity problem (pp. 558–569). Google Scholar
  2. April, J., Glover, F., Kelly, J. P., & Laguna, M. (2003). Simulation-based optimization: practical introduction to simulation optimization. In Proceedings of the 35th conference on Winter simulation: driving innovation (pp. 71–78). Google Scholar
  3. Aringhieri, R., & Cordone, R. (2011). Comparing local search metaheuristics for the maximum diversity problem. The Journal of the Operational Research Society, 62(2), 266–280. CrossRefGoogle Scholar
  4. Aringhieri, R., Cordone, R., & Melzani, Y. (2008). Tabu search versus GRASP for the maximum diversity problem. 4OR: A Quarterly Journal of Operations Research, 6(1), 45–60. CrossRefGoogle Scholar
  5. Della Croce, F., Grosso, A., & Locatelli, M. (2009). A heuristic approach for the max-min diversity problem based on max-clique. Computers and Operations Research, 36(8), 2429–2433. CrossRefGoogle Scholar
  6. Duarte, A., & Marti, R. (2007). Tabu search and GRASP for the maximum diversity problem. European Journal of Operational Research, 178, 71–84. CrossRefGoogle Scholar
  7. Erkut, E. (1990). The discrete dispersion problem. European Journal of Operational Research, 46, 48–60. CrossRefGoogle Scholar
  8. Gallego, M., Duarte, A., Laguna, M., & Marti, R. (2009). Hybrid heuristics for the maximum diversity problem. Computational Optimization and Applications, 44(3), 411–426. CrossRefGoogle Scholar
  9. Ghosh, J. B. (1996). Computational aspects of the maximum diversity problem. Operations Research Letters, 19, 175–181. CrossRefGoogle Scholar
  10. Glover, F., & Laguna, M. (1997). Tabu search. Dordrecht: Kluwer Academic. Google Scholar
  11. Glover, F., Kuo, C., & Dhir, K. (1998). Heuristic algorithms for the maximum diversity problem. Journal of Information and Optimization Sciences, 19(1), 109–132. Google Scholar
  12. Grosso, A., Locatelli, M., & Pullan, W. (2008). Randomness, plateau search, penalties, restart rules: simple ingredients leading to very efficient heuristics for the maximum clique problem. Journal of Heuristics, 14(6), 587–612. CrossRefGoogle Scholar
  13. Kincaid, R. (1992). Good solutions to discrete noxious location problems via metaheuristics. Annals of Operation Research, 40, 265–281. CrossRefGoogle Scholar
  14. Kuo, C., Glover, F., & Dhir, K. (1993). Analyzing and modeling the maximum diversity problem by zero-one programming. Decision Sciences, 24(6), 1171–1185. CrossRefGoogle Scholar
  15. Östergård, P. R. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120(1–3), 197–207. CrossRefGoogle Scholar
  16. Palubeckis, G. (2007). Iterated tabu search for the maximum diversity problem. Applied Mathematics and Computation, 189(1), 371–383. CrossRefGoogle Scholar
  17. Resende, M., Marti, R., Gallego, M., & Duarte, A. (2010). GRASP and path relinking for the max-min diversity problem. Computers and Operations Research, 37(3), 498–408. CrossRefGoogle Scholar
  18. Santos, L., Martins, S., & Plastino, A. (2008). Applications of the DM-GRASP heuristic: a survey. International Transactions in Operational Research, 15(4), 387–416. CrossRefGoogle Scholar
  19. Silva, G. C., Ochi, L. S., & Martins, S. L. (2004). Lecture notes in computer science: Vol. 3059. Experimental comparison of greedy randomized adaptive search procedures for the maximum diversity problem (pp. 498–512). Google Scholar
  20. Wang, J., Zhou, Y., Yin, J., & Zhang, Y. (2009). Competitive hopfield network combined with estimation of distribution for maximum diversity problems. IEEE Transactions on Systems, Man and Cybernetics. Part B. Cybernetics, 39(4), 1048–1066. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Daniel Cosmin Porumbel
    • 1
  • Jin-Kao Hao
    • 2
  • Fred Glover
    • 3
  1. 1.UArtois, LGI2AUniv. Lille-Nord de FranceBéthuneFrance
  2. 2.LERIAUniversité d’AngersAngers Cedex 01France
  3. 3.OptTek Systems, Inc.BoulderUSA

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