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

Abstract

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.

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Correspondence to Ellips Masehian.

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Masehian, E., Akbaripour, H. & Mohabbati-Kalejahi, N. Landscape analysis and efficient metaheuristics for solving the n-queens problem. Comput Optim Appl 56, 735–764 (2013). https://doi.org/10.1007/s10589-013-9578-z

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Keywords

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