Computational Optimization and Applications

, Volume 56, Issue 3, pp 735–764 | Cite as

Landscape analysis and efficient metaheuristics for solving the n-queens problem

  • Ellips Masehian
  • Hossein Akbaripour
  • Nasrin Mohabbati-Kalejahi


The n-queens problem is a classical combinatorial optimization problem which has been proved to be NP-hard. The goal is to place n non-attacking queens on an n×n chessboard. In this paper, two single-solution-based (Local Search (LS) and Tuned Simulated Annealing (SA)) and two population-based metaheuristics (two versions of Scatter Search (SS)) are presented for solving the problem. Since parameters of heuristic and metaheuristic algorithms have great influence on their performance, a TOPSIS-Taguchi based parameter tuning method is proposed, which not only considers the number of fitness function evaluations, but also aims to minimize the runtime of the presented metaheuristics. The performance of the suggested approaches was investigated through computational analyses, which showed that the Local Search method outperformed the other two algorithms in terms of average runtimes and average number of fitness function evaluations. The LS was also compared to the Cooperative PSO (CPSO) and SA algorithms, which are currently the best algorithms in the literature for finding the first solution to the n-queens problem, and the results showed that the average fitness function evaluation of the LS is approximately 21 and 8 times less than that of SA and CPSO, respectively. Also, a fitness analysis of landscape for the n-queens problem was conducted which indicated that the distribution of local optima is uniformly random and scattered over the search space. The landscape is rugged and there is no significant correlation between fitness and distance of solutions, and so a local search heuristic can search a rugged plain landscape effectively and find a solution quickly. As a result, it was statistically and analytically proved that single-solution-based metaheuristics outperform population-based metaheuristics in finding the first solution of the n-queens problem.


n-Queens problem Local search Simulated annealing Scatter search Parameter tuning TOPSIS method Fitness analysis of landscape 


  1. 1.
    Aarts, E.H.L., Lenstra, J.K.: Local Search in Combinatorial Optimization. Wiley, New York (1997) MATHGoogle Scholar
  2. 2.
    Abramson, B., Yung, M.: Divide and conquer under global constraints: a solution to the N-queens problem. J. Parallel Distrib. Comput. 6(3), 649–662 (1989) CrossRefGoogle Scholar
  3. 3.
    Amooshahi, A., Joudaki, M., Imani, M., Mazhari, N.: Presenting a new method based on cooperative PSO to solve permutation problems: a case study of n-queen problem. In: 3rd International Conference on Electronics Computer Technology (2011) Google Scholar
  4. 4.
    Bell, J., Stevens, B.: A survey of known results and research areas for n-queens. Discrete Math. 309, 1–31 (2009) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bezzel, M.: Proposal of 8-queens problem, Berliner Schachzeitung 3, 363 (1848) Google Scholar
  6. 6.
    Campos, V., Laguna, M., Mart, R.: Context-independent scatter search and tabu search for permutation problems. INFORMS J. Comput. 17, 111–122 (2005) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dirakkhunakon, S., Suansook, Y.: Simulated annealing with iterative improvement. In: International Conference on Signal Processing Systems (2009). doi: 10.1109/ICSPS.2009.61 Google Scholar
  8. 8.
    Draa, A., Talbi, H., Batouche, M.: A quantum inspired genetic algorithm for solving the N-queens problem. In: Proceedings of the 7th International Symposium on Programming and Systems, Algiers, pp. 145–152 (2005) Google Scholar
  9. 9.
    Draa, A., Meshoul, S., Talbi, H., Batouche, M.: A quantum-inspired differential evolution algorithm for solving the N-queens problem. Int. Arab J. Inf. Technol. 7, 21–27 (2010) Google Scholar
  10. 10.
    Erbas, C., Sarkeshik, S., Tanik, M.M.: Different perspectives of the n-queens problem. In: Proceedings of the 1992 ACM Annual Conference on Communications, pp. 99–108. ACM Press, New York (1992) CrossRefGoogle Scholar
  11. 11.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984) CrossRefMATHGoogle Scholar
  12. 12.
    Glover, F.: A template for scatter search and path relinking. In: Hao, J.K., Lutton, E., Ronald, E., Schoenauer, D., Snyers, D. (eds.) Lecture Notes in Computer Science, vol. 1363, pp. 13–54 (1997) Google Scholar
  13. 13.
    Glover, F.: Heuristics for integer programming using surrogate constraints. Decis. Sci. 8, 156–166 (1977) CrossRefGoogle Scholar
  14. 14.
    Homaifar, A., Turner, J., Ali, S.: The n-queens problem and genetic algorithms. In: Proceedings IEEE Southeast Conference, vol. 1, pp. 262–267 (1992) CrossRefGoogle Scholar
  15. 15.
    Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making-Method and Applications, a State-of-the-Art Survey. Springer, New York (1981) CrossRefGoogle Scholar
  16. 16.
    Jagota, A.: Optimization by reduction to maximum clique. In: IEEE International Conference on Neural Networks (1993) Google Scholar
  17. 17.
    Jones, T., Forrest, S.: Fitness distance correlation as a measure of problem difficulty for genetic algorithms. In: Proceedings of the 6th International Conference on GeneticAlgorithms, pp. 184–192. Morgan Kaufmann, San Francisco (1995) Google Scholar
  18. 18.
    Jones, T.: Evolutionary algorithms, fitness landscapes and search. PhD thesis, University of New Mexico, Albuquerque, NM (1995) Google Scholar
  19. 19.
    Kale, L.V.: An almost perfect heuristic for the n non-attacking queens problem. Inf. Process. Lett. 34, 173–178 (1990) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Khan, S., Bilal, M., Sharif, M., Sajid, M., Baig, R.: Solution of n-queen problem using ACO. In: IEEE 13th International Multitopic Conference, Art. No. 5383157 (2009). Google Scholar
  21. 21.
    Kirkpatrick, S., Gelet, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 621–630 (1983) CrossRefGoogle Scholar
  22. 22.
    Kosters, W.: n-queens bibliography. Retrieved May 4 (2012).
  23. 23.
    Laguna, M., Marti, R.: Scatter Search: Methodology and Implementations in C. Kluwer Academic, Boston (2003) CrossRefGoogle Scholar
  24. 24.
    Lionnet, F.J.E.: Question 963. Nouv. Ann. Math., Ser. 2 8, 560 (1869) Google Scholar
  25. 25.
    Martí, R., Laguna, M., Campos, V.: Scatter search vs. genetic algorithms: an experimental evaluation with permutation problems. In: Rego, C., Alidaee, B. (eds.) Adaptive Memory and Evolution: Tabu Search and Scatter Search. Kluwer Academic, Dordrecht (2004) Google Scholar
  26. 26.
    Martinjak, I., Golub, M.: Comparison of heuristic algorithms for the N-queen problem. In: Proceedings of the ITI 2007 29th Int. Conf. on Information Technology Interfaces, Cavtat, Croatia, pp. 25–28 (2007) Google Scholar
  27. 27.
    Nouraniy, Y., Andresenz, B.: A comparison of simulated annealing cooling strategies. J. Phys. A, Math. Gen. 31, 8373–8385 (1998) CrossRefGoogle Scholar
  28. 28.
    Pauls, E.: Das Maximalproblem der Damen auf dem Schachbrete, II, Deutsche Schachzeitung. Organ fur das Gesammte Schachleben 29(9), 257–267 (1874) Google Scholar
  29. 29.
    Rego, C., Leão, P.: A scatter search tutorial for graph-based permutation problems. Research Paper HCES-10-00, Hearin Center for Enterprise Science, University of Mississippi, MS 38677, USA (2009) Google Scholar
  30. 30.
    Rivin, I., Zabih, R.: A dynamic programming solution to the n-queens problem. Inf. Process. Lett. 41, 253–256 (1992) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Russell, S.J., Norvig, P.: Artificial Intelligence a Modern Approach Prentice-Hall, Englewood Cliffs (1995) MATHGoogle Scholar
  32. 32.
    Segundo, P.S.: New decision rules for exact search in N-queens. J. Glob. Optim. 51, 497–514 (2011). doi: 10.1007/s10898-011-9653-x CrossRefMATHGoogle Scholar
  33. 33.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2012).
  34. 34.
    Sosic, R., Gu, J.: Efficient local search with conflict minimization. IEEE Trans. Knowl. Data Eng. 6E, 661–668 (1994) CrossRefGoogle Scholar
  35. 35.
    Taguchi, G., Yokoyama, Y.: Taguchi Methods: Design of Experiments. Am. Supplier Inst. Press, Millersburg (1993) Google Scholar
  36. 36.
    Talbi, E.-G.: Metaheuristics from Design to Implementation. Wiley, Hoboken (2009) MATHGoogle Scholar
  37. 37.
    Tambouratzis, T.: A simulated annealing artificial neural network implementation of the n-queens problem. Int. J. Intell. Syst. 12, 739–752 (1997) CrossRefGoogle Scholar
  38. 38.
    Tong, L.I., Wang, Ch.H., Chen, H.C.: Optimization of multiple responses using principal component analysis and technique for order preference by similarity to ideal solution. Int. J. Adv. Manuf. Technol. 27, 407–414 (2005) CrossRefGoogle Scholar
  39. 39.
    Wolpert, D.W., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1(1), 67–82 (1997) CrossRefGoogle Scholar
  40. 40.
    Xinchao, Z.: Simulated annealing algorithm with adaptive neighborhood. Appl. Soft Comput. 11, 1827–1836 (2011) CrossRefGoogle Scholar
  41. 41.
    Yang, X.-S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press (2010) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ellips Masehian
    • 1
  • Hossein Akbaripour
    • 1
  • Nasrin Mohabbati-Kalejahi
    • 2
  1. 1.Industrial Engineering DepartmentTarbiat Modares UniversityTehranIran
  2. 2.Faculty of Industrial EngineeringAmirkabir University of TechnologyGarmsar CampusIran

Personalised recommendations