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Generating quadratic assignment test problems with known optimal permutations

Abstract

The quadratic assignment problem (QAP) is a well-known combinatorial optimization problem of which the travelling-salesman problem is a special case. Although the QAP has been extensively studied during the past three decades, this problem remains very hard to solve. Problems of sizes greater than 15 are generally impractical to solve. For this reason, many heuristics have been developed. However, in the literature, there is a lack of test problems with known optimal solutions for evaluating heuristic algorithms. Only recently Paulubetskis proposed a method to generate test problems with known optimal solutions for a special type of QAP. This paper concerns the generation of test problems for the QAP with known optimal permutations. We generalize the result of Palubetskis and provide test-problem generators for more general types of QAPs. The test-problem generators proposed are easy to implement and were also tested on several well-known heuristic algorithms for the QAP. Computatinal results indicate that the test problems generated can be used to test the effectiveness of heuristic algorithms for the QAP. Comparison with Palubetskis' procedure was made, showing the superiority of the new test-problem generators. Three illustrative test problems of different types are also provided in an appendix, together with the optimal permutations and the optimal objective function values.

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References

  1. 1.

    M.S. Bazaraa and H.D. Sherali, “On the use of exact and heuristic cutting plane methods for the quadratic assignment problem,” J. of Oper. Res., vol. 33, pp. 991–1003, 1982.

  2. 2.

    R.E. Burkard, Locations with spatial interactions: The quadratic assignment problem, “Discrete location theory,” Chapter 9, (P.B. Mirchandani and R.L. Francis, eds.), John Wiley & Sons, Inc., Berlin, 1990.

  3. 3.

    R.E. Burkard and F. Rendl, “A thermodynamically motivated simulation procedure for combinatorial optimization problems,” European J. of Oper. Res., vol. 17, pp. 169–174, 1984.

  4. 4.

    C.A. Floudas and P.M. Pardalos. A collection of test problems for constrained global optimization algorithms, Lecture Notes in Computer Science, No. 455, Springer-Verlag, 1990.

  5. 5.

    P.C. Gilmore, “Optimal and suboptimal algorithms for the quadratic assignment program,” J. SIAM, vol. 10, pp. 305–313, 1962.

  6. 6.

    G.G. Hardy, J.E. Littlewood, and G. Polya. Inequalities, Cambridge University Press: London, 1952.

  7. 7.

    T.C. Koopmans and M.J. Beckmann, “Assignment problems and the location of economic activities”, Econometrica, vol. 25, pp. 53–76, 1957.

  8. 8.

    E.L. Lawler. “The quadratic assignment problem,” Management. Sci., vol. 9, pp. 586–599, 1963.

  9. 9.

    Y. Li, P.M. Pardalos K.G. Ramakrishnan, and M.G.C. Resende. “Lower bounds for the quadratic assignment problem,” submitted to Annals of Oper. Res., 1992.

  10. 10.

    K.A. Murthy and P.M. Pardalos, “A polynomial-time approximation algorithm for the quadratic assignment problem,” The Pennsylvania State University, Technical Report CS-33-90, 1990.

  11. 11.

    K.A. Murthy, P.M. Pardalos, and Y. Li “A local search algorithm for the quadratic assignment problem,” submitted to J. of Global Optimization, 1992.

  12. 12.

    C.E. Nugent, T.E. Vollmann, and J. Ruml, “An experimental comparison of techniques for the assignment of facilities to locations,” J. of Oper. Res., vol. 16, pp. 150–173, 1969.

  13. 13.

    G.S. Palubetskis, “Generation of quadratic assignment test problems with known optimal solutions (in Russian),” Zh. Vychisl. Mat. Mat. Fiz., vol. 28, pp. 1740–1743, 1988.

  14. 14.

    P.M. Pardalos, “Generation of large-scale quadratic programs for use as global optimization test problems,” ACM Trans. on Math. Software, vol. 13, pp. 133–137, 1987.

  15. 15.

    P.M. Pardalos, “Construction of test problems in quadratic bivalent programming,” ACM Trans. on Math., vol. 17, pp. 74–87, 1991.

  16. 16.

    P.M. Pardalos and J. Crouse, “A parallel algorithm for the quadratic assignment problem,” in Proc. of the Supercomputing 1989 Conf., ACM Press, pp. 351–360, 1989.

  17. 17.

    P.M. Pardalos, K.A. Murthy, and Y. Li, “Computational experience with parallel algorithms for solving the quadratic assignment problem,” appear in Computer Science and Operations Research: New developments in their interfaces, Williamsburg, VA, 1992. ORSA CSTS.

  18. 18.

    P.M. Pardalos and J.B. Rosen, Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science, No. 268, Springer-Verlag: 1987.

  19. 19.

    J. Skorin-Kapov, “Tabu search applied to the quadratic assignment problem,” ORSA J. on Computing, vol. 2, pp. 33–45, 1990.

  20. 20.

    L. Steinberg, “The backboard wiring problem: A placement algorithm,” Siam Review; vol. 3, pp. 37–50, 1961.

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Li, Y., Pardalos, P.M. Generating quadratic assignment test problems with known optimal permutations. Comput Optim Applic 1, 163–184 (1992). https://doi.org/10.1007/BF00253805

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Keywords

  • test problems
  • quadratic assignment
  • heuristics
  • combinatorial optimization