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Mathematical Programming

, 122:225 | Cite as

Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem

  • Etienne de Klerk
  • Renata Sotirov
Open Access
FULL LENGTH PAPER Series A

Abstract

We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: (Burkard et al. in J Global Optim 10:291–403, 1997).

Keywords

Quadratic assignment problem Semidefinite programming Group symmetry 

Mathematics Subject Classification (2000)

90C22 20Cxx 70-08 

Notes

Acknowledgments

The authors are very grateful to Dima Pasechnik for suggesting the heuristic in Sect. 5.2.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Anstreicher K.M.: Recent advances in the solution of quadratic assignment problems. Math. Program. Ser. B 97, 27–42 (2003)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Anstreicher K.M., Brixius N., Linderoth J., Goux J.-P.: Solving large quadratic assignment problems on computational grids. Math. Program. Ser. B 91, 563–588 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borchers B.: CSDP, a C library for semidefinite programming. Optim. Meth. Softw. 11/12(1–4), 613–623 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Burer S., Vandenbussche D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eschermann, B., Wunderlich, H.J.: Optimized synthesis of self-testable finite state machines. In: 20th International Symposium on Fault-tolerant Computing (FFTCS 20), Newcastle upon Tyne, 26–28 June 1990Google Scholar
  6. 6.
    Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB—a quadratic assignment problem library. J. Global Optim. 10, 291–403 (1997); see also http://www.seas.upenn.edu/qaplib/ Google Scholar
  7. 7.
    Klerk E., Maharry J., Pasechnik D.V., Richter B., Salazar G.: Improved bounds for the crossing numbers of K m,n and K n. SIAM J. Discr. Math. 20, 189–202 (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    Klerk E., Pasechnik D.V., Schrijver A.: Reduction of symmetric semidefinite programs using the regular *-representation. Math. Program. B 109(2–3), 613–624 (2007)zbMATHCrossRefGoogle Scholar
  9. 9.
    de Klerk, E., Pasechnik, D.V., Sotirov, R.: On semidefinite programming relaxations of the traveling salesman problem. CentER discussion paper 2007-101, Tilburg University, The Netherlands (2007). Available at: http://arno.uvt.nl/show.cgi?fid=67481
  10. 10.
    GAP—groups, algorithms, and programming, Version 4.4.9, The GAP group (2006) http://www.gap-system.org
  11. 11.
    Gatermann K., Parrilo P.A.: Symmetry groups, semidefinite programs, and sum of squares. J. Pure Appl. Algebra 192, 95–128 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gijswijt, D.: Matrix algebras and semidefinite programming techniques for codes. PhD Thesis, University of Amsterdam, The Netherlands (2005). http://staff.science.uva.nl/~gijswijt/promotie/thesis.pdf
  13. 13.
    Gilmore P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10, 305–31 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Graham A.: Kronecker products and matrix calculus with applications. Ellis Horwood Limited, Chichester (1981)zbMATHGoogle Scholar
  15. 15.
    Grone R., Johnson C.R., Sá E.M., Wolkowicz H.: Positive definite completions of partial Hermitian matrices. Lin. Algebra Appl. 58, 109–124 (1984)zbMATHCrossRefGoogle Scholar
  16. 16.
    Lawler E.: The quadratic assignment problem. Manage. Sci. 9, 586–599 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lovász L., Schrijver A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Peng, J., Mittelmann, H., Wu, X.: Graph modeling for quadratic assignment problem associated with the hypercube. Technical report, Department of IESE, UIUC, Urbana, IL, 61801, USA (2007). Available at Optimization OnlineGoogle Scholar
  19. 19.
    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Manuscript (2006). http://www.optimization-online.org/DB_HTML/2006/10/1502.html
  20. 20.
    Rendl F., Sotirov R.: Bounds for the quadratic assignment problem using the bundle method. Math Program Ser B 109(2–3), 505–524 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schrijver A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory 25, 425–429 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Schrijver A.: New code upper bounds from the Terwilliger algebra. IEEE Trans Inf Theory 51, 2859–2866 (2005)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Serre J.-P.: Linear representations of finite groups. Graduate texts in mathematics, vol. 42. Springer, New York (1977)Google Scholar
  24. 24.
    Sturm J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Meth. Softw. 11–12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Wedderburn J.H.M.: On hypercomplex numbers. Proc. Lond. Math. Soc. 6(2), 77–118 (1907)Google Scholar
  26. 26.
    Wedderburn, J.H.M.: Lectures on matrices. AMS Colloquium Publications, vol. 17. AMS publishers (1934)Google Scholar
  27. 27.
    Löfberg, J.: YALMIP : A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD conference, Taipei, Taiwan (2004). http://control.ee.ethz.ch/~joloef/yalmip.php
  28. 28.
    Zhao Q., Karisch S.E., Rendl F., Wolkowicz H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Combin. Optim. 2, 71–109 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Department of Econometrics and ORTilburg UniversityTilburgThe Netherlands

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