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Lower bounds for the quadratic assignment problem

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Abstract

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.

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References

  1. A. Assad and W. Xu, On lower bounds for a class of quadratic {0,1} programs, Oper. Res. Lett. 4(1985)175–180.

    Google Scholar 

  2. R. Burkard, Die Störungsmethode zur Lösung quadratischer Zuordnungsprobleme, Oper. Res. Verfahren 16(1973)84–108.

    Google Scholar 

  3. R. Burkard, Locations with spatial interactions: the quadratic assignment problem, in:Discrete Location Theory, ed. P.B. Mirchandani and R.L. Francis (Wiley, Berlin, 1990) chapter 9.

    Google Scholar 

  4. R. Burkard and T. Bonniger, A heuristic for quadratic Boolean programs with applications to quadratic assignment problems, Euro. J. Oper. Res. 13(1983)374–386.

    Google Scholar 

  5. R. Burkard and F. Rendl, A thermodynamically motivated simulation procedure for combinatorial optimization problems, Euro. J. Oper. Res. 17(1984)169–174.

    Google Scholar 

  6. R. Burkard and K. Stratmann, Numerical investigations on quadratic assignment problems, Naval Res. Log. Quarterly 25(1978)129–148.

    Google Scholar 

  7. R. Burkard and J. Offerman, Entwurf von Schreibmachinentastaturen mittels quadratischer Zuordnungsprobleme, Z. Oper. Res. 21(1977)B121-B132.

    Google Scholar 

  8. P. Carraresi and F. Malucelli, A new lower bound for the quadratic assignment problem. Oper. Res. 40(1992)S22-S27.

    Google Scholar 

  9. N. Christofides and M. Gerrard, A graph theoretic analysis of bounds for the quadratic assignment problem, in:Studies on Graphs and Discrete Programming, ed. P. Hansen (North-Holland, 1981) pp. 61–68.

  10. C. Edwards, A branch and bound algorithm for the Koopmans-Beckman quadratic assignment problem, Math. Progr. Study 13(1980)35–52.

    Google Scholar 

  11. G. Finke, R. Burkard and F. Rendl, Quadratic assignment problems, Ann. Discr. Math. 31(1987)61–82.

    Google Scholar 

  12. A. Frieze and J. Yadegar, On the quadratic assignment problem, Discr. Appl. Math, 5(1983)89–98.

    Google Scholar 

  13. M. Garey and D. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman, New York, 1979).

    Google Scholar 

  14. P. Gilmore, Optimal and suboptimal algorithms for the quadratic assignment problem. J. SIAM 10(1962)305–313.

    Google Scholar 

  15. G. Gordon,System Simulation (Prentice-Hall, Englewood Cliffs, NJ, 1969).

    Google Scholar 

  16. S. Hadley, F. Rendl and H. Wolkowicz, Bounds for the quadratic assignment problem using continuous optimization tehcniques, in:Integer Programming and Combinatorial Optimization (University of Waterloo Press, 1990) pp. 237–248.

  17. S. Hadley, F. Rendl and H. Wolkowicz, A new lower bound via projection for the quadratic assignment problem, Math. Oper. Res. 17(1992)727–739.

    Google Scholar 

  18. T. Koopmans and M. Beckmann, Assigned problems and the location of economic activities, Econometrica 25(1957)53–76.

    Google Scholar 

  19. J. Krarup and P. Pruzan, Computer-aided layout design, Math. Progr. Study 9(1978)75–94.

    Google Scholar 

  20. E. Lawler, The quadratic assignment problem, Manag. Sci. 9(1963)586–599.

    Google Scholar 

  21. Y. Li, P. Pardalos, K. Ramakrishnan and M. Resende, A branch-and-bound algorithm for the quadratic assignment problem, Technical Report, AT&T Bell Laboratories, Murray Hill, NJ 07974-2070 (December 1992).

    Google Scholar 

  22. Y. Li and P.M. Pardalos, Generating quadratic assignment test problems with known optimal permutations, Comp. Optim. Appl. (1992), to appear.

  23. Y. Li and P.M. Pardalos, Parallel algorithms for the quadratic assignment problem, in:Recent Advances in Optimization and Parallel Computing (Elsevier, Amsterdam, 1922) pp. 177–189.

    Google Scholar 

  24. K. Murthy, P. Pardalos and Y. Li, A local search algorithm for the quadratic assignment problem, Informatica 3(1992).

  25. C. Nugent, T. Vollmann and J. Ruml, An experimental comparison of techniques for the assignment of facilities to locations, J. Oper. Res. 16(1969)150–173.

    Google Scholar 

  26. L. Nyhoff and S. Leestma,Fortran 77 for Engineers and Scientist, 2nd ed. (Macmillan, New York, 1988).

    Google Scholar 

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

    Google Scholar 

  28. P. Pardalos and J. Crouse, A parallel algorithm for the quadratic assignment problem, in:Proc. Supercomputing 1989 Conf. (ACM Press, 1989) pp. 351–360.

  29. P. Paradalos and G. Rodgers, Parallel branch and bound algorithms for unconstrained quadratic 0–1 programming, in:Impact of Recent Advances on Operations Research (North-Holland, 1989) pp. 131–143.

  30. P. Pardalos and J. Rosen,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science, No. 268 (Springer, 1987).

  31. F. Rendl and H. Wolkowicz, Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem, Math. Progr. 53(1992)63–78.

    Google Scholar 

  32. C. Roucairol, Affection quadratique, Ph.D. Thesis, Université Pierre et Marie Curie, Paris VI (1976).

    Google Scholar 

  33. C. Roucairol, A reduction method for quadratic assignment problems, Oper. Res. Verfahren 32(1979)183–187.

    Google Scholar 

  34. C. Roucairol, Un nouvel algorithme pour le problème d'affectation quadratique, RAIRO 13(1979)275–301.

    Google Scholar 

  35. S. Sahni and T. Gonzalez, P-complete approximation problems, J. ACM 23(1976)555–565.

    Google Scholar 

  36. L. Schrage, A more portable Fortran random number generator, ACM Trans. Math. Software 5(1979)132–138.

    Google Scholar 

  37. J. Skorin-Kapov, Tabu search applied to the quadratic assignment problem, ORSA J. Comp. 2(1990)33–45.

    Google Scholar 

  38. E. Taillard, Robust tabu search for the quadratic assignment problem, Parallel Comp. 17(1991)443–455.

    Google Scholar 

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Authors and Affiliations

  1. Computer Science Department, The Pennsylvania State University, 16802, University Park, PA, USA

    Y. Li

  2. Department of Industrial and Systems Engineering, University of Florida, 32611, Gainesville, FL, USA

    P. M. Pardalos

  3. Mathematical Sciences Research Center, AT&T Bell Laboratories, 07974, Murray Hill, NJ, USA

    K. G. Ramakrishnan & M. G. C. Resende

Authors
  1. Y. Li
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  2. P. M. Pardalos
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  3. K. G. Ramakrishnan
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  4. M. G. C. Resende
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Li, Y., Pardalos, P.M., Ramakrishnan, K.G. et al. Lower bounds for the quadratic assignment problem. Ann Oper Res 50, 387–410 (1994). https://doi.org/10.1007/BF02085649

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