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On the Applicability of Lower Bounds for Solving Rectilinear Quadratic Assignment Problems in Parallel

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Abstract

The quadratic assignment problem (QAP) belongs to the hard core of NP-hard optimization problems. After almost forty years of research only relatively small instances can be solved to optimality. The reason is that the quality of the lower bounds available for exact methods is not sufficient. Recently, lower bounds based on decomposition were proposed for the so called rectilinear QAP that proved to be the strongest for a large class of problem instances. We investigate the strength of these bounds when applied not only at the root node of a search tree but as the bound function used in a Branch-and-Bound code solving large scale QAPs.

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References

  1. W.P. Adams and T.A. Johnson, “Improved linear programming-based lower bounds for the quadratic assignment problem,” in Quadratic Assignment and Related Problems, P. Pardalos and H. Wolkowicz (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1994, vol. 16, pp. 43-75.

  2. A.A. Assad and W. Xu, “On lower bounds for a class of quadratic {0, 1} programs,” Oper. Res. Letters, vol. 4, pp. 175-180, 1985.

    Google Scholar 

  3. M.S. Bazaraa and O. Kirca, “A branch-and-bound-based heuristic for solving the quadratic assignment problem,” Naval Research Logistics Quarterly, vol. 30, pp. 287-304, 1983.

    Google Scholar 

  4. A. Brüngger, J. Clausen, A. Marzetta, and M. Perregaard, “Joining forces in solving large-scale quadratic assignment problems in parallel,” Proceedings of IPPS' 97, 11, IEEE International Parallel Processing Symposium, pp. 418-427, 1997.

  5. R.E. Burkard, “Locations with spatial interactions: The quadratic assignment problem,” in Discrete Location Theory, P.B. Mirchandani and R.L. Francis (Eds.), Wiley: Berlin, 1991.

    Google Scholar 

  6. R.E. Burkard and F. Rendl, “A thermodynamically motivated simulation procedure for combinatorial optimization problems,” European Journal of Operations Research, vol. 17, no.2, pp. 169-174, 1984.

    Google Scholar 

  7. R.E. Burkard and E. Çela, "Quadratic and three-dimensional assignment problems," in Annotated Bibliographies in Combinatorial Optimization, M. Dell'Amico, F. Maffioli, and S. Martello (Eds.), Wiley, Chichester, pp. 373-392, 1997.

    Google Scholar 

  8. R.E. Burkard, S.E. Karisch, and F. Rendl, "QAPLIB-A quadratic assignment problem library," Journal of Global Optimization, vol. 10, pp. 391-403, 1997. QAPLIB Home Page: http://www.diku.dk/~karisch/qaplib.

    Google Scholar 

  9. P. Carraresi and F. Malucelli, "A new lower bounds for the quadratic assignment problem," Operations Research, vol. 40,Supplement 1:S22-S27, 1992.

    Google Scholar 

  10. P. Carraresi and F. Malucelli, "A reformulation scheme and new lower bounds for the QAP,” in Quadratic Assignment and Related Problems, P. Pardalos and H. Wolkowicz (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1994, vol. 16, pp. 147-160.

  11. J. Chakrapani and J. Skorin-Kapov, "A constructive method to improve lower bounds for the quadratic assignment problem,” in Quadratic Assignment and Related Problems, P. Pardalos and H. Wolkowicz (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1994, vol. 16, pp. 161-171.

  12. J. Clausen and M. Perregaard, "Solving large quadratic assignment problems in parallel," Computational Optimization and Applications, vol. 8, pp. 111-128, 1997.

    Google Scholar 

  13. G. Finke, R.E. Burkard, and F. Rendl, "Quadratic assignment problems," Annals of Discrete Mathematics, vol. 31, pp. 61-82, 1987.

    Google Scholar 

  14. P.C. Gilmore, "Optimal and suboptimal algorithms for the quadratic assignment problem," SIAM Journal on Applied Mathematics, vol. 10, pp. 305-331, 1962.

    Google Scholar 

  15. S.W. Hadley, F. Rendl, and H. Wolkowicz, "A new lower bound via projection for the quadratic assignment problem," Mathematics of Operations Research, vol. 17, pp. 727-739, 1992.

    Google Scholar 

  16. P. Hahn, T. Grant, and N. Hall, "Solution of the quadratic assignment problem using the Hungarian method," European Journal of Operational Research, 1995, to appear.

  17. S.E. Karisch, “Nonlinear approaches for the quadratic assignment and graph partition problems,” Ph.D. thesis, Graz University of Technology, Graz, Austria, 1995.

    Google Scholar 

  18. S.E. Karisch and F. Rendl, "Lower bounds for the quadratic assignment problem via triangle decompositions," Mathematical Programming, vol. 71, no.2, pp. 137-152, 1995.

    Google Scholar 

  19. T.C. Koopmans and M.J. Beckmann, "Assignment problems and the location of economic activities," Econometrica, vol. 25, pp. 53-76, 1957.

    Google Scholar 

  20. E. Lawler, "The quadratic assignment problem," Management Science, vol. 9, pp. 586-599, 1963.

    Google Scholar 

  21. T. Mautor and C. Roucairol, "A new exact algorithm for the solution of quadratic assignment problems," Discrete Applied Mathematics, vol. 55, pp. 281-293, 1994.

    Google Scholar 

  22. C.E. Nugent, T.E. Vollman, and J. Ruml, "An experimental comparison of techniques for the assignment of facilities to locations," Operations Research, vol. 16, pp. 150-173, 1968.

    Google Scholar 

  23. P.M. Pardalos, F. Rendl, and H. Wolkowicz, "The quadratic assignment problem: A survey of recent developments,” in Quadratic Assignment and Related Problems, P. Pardalos and H. Wolkowicz (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1994, vol. 16, pp. 1-42.

  24. F. Rendl and H. Wolkowicz, "Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem," Mathematical Programming, vol. 53, pp. 63-78, 1992.

    Google Scholar 

  25. S. Sahni and T. Gonzales, "P-complete approximation problems," Journal of the Association of Computing Machinery, vol. 23, pp. 555-565, 1976.

    Google Scholar 

  26. H. Schramm and J. Zowe, "A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results," SIAM Journal of Optimization, vol. 2, pp. 121-152, 1992.

    Google Scholar 

  27. H. Schramm and J. Zowe, "A combination of the bundle approach and the trust region concept," in Advances in Mathematical Optimization, J. Guddat et al. (Eds.), Akademie Verlag Berlin, 1994.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100, Copenhagen, Denmark

    Jens Clausen, Stefan E. Karisch & Michael Perregaard

  2. Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010, Graz, Austria

    Franz Rendl

Authors
  1. Jens Clausen
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  2. Stefan E. Karisch
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  3. Michael Perregaard
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  4. Franz Rendl
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Clausen, J., Karisch, S.E., Perregaard, M. et al. On the Applicability of Lower Bounds for Solving Rectilinear Quadratic Assignment Problems in Parallel. Computational Optimization and Applications 10, 127–147 (1998). https://doi.org/10.1023/A:1018308718386

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