A Novel SDP Relaxation for the Quadratic Assignment Problem Using Cut Pseudo Bases

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9849)


The quadratic assignment problem (QAP) is one of the hardest combinatorial optimization problems. Its range of applications is wide, including facility location, keyboard layout, and various other domains. The key success factor of specialized branch-and-bound frameworks for minimizing QAPs is an efficient implementation of a strong lower bound. In this paper, we propose a lower-bound-preserving transformation of a QAP to a different quadratic problem that allows for small and efficiently solvable SDP relaxations. This transformation is self-tightening in a branch-and-bound process.


Quadratic assignment Semidefinite program Lower bound Branch and bound 


  1. 1.
    Birkhoff, D.: Tres observaciones sobre el algebra lineal. Univ. Nac. Tucuman Rev. Ser. A 5, 147–151 (1946)MathSciNetGoogle Scholar
  2. 2.
    Kuhn, H.W.: The hungarian method for the assignment problem. Naval Res. Logistics Q. 2, 83–97 (1955)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Duan, R., Su, H.H.: A scaling algorithm for maximum weight matching in bipartite graphs. In: Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1413–1424. SIAM (2012)Google Scholar
  4. 4.
    Koopmans, T., Beckmann, M.J.: Assignment problems and the location of economic activities. Cowles Foundation Discussion Papers 4, Cowles Foundation for Research in Economics, Yale University (1955)Google Scholar
  5. 5.
    Nugent, C., Vollman, T., Ruml, J.: An experimental comparison of techniques for the assignment of facilities to locations. Oper. Res. 16, 150–173 (1968)CrossRefGoogle Scholar
  6. 6.
    Burkard, R., Offermann, J.: Entwurf von Schreibmaschinentastaturen mittels quadratischer Zuordnungsprobleme. Z. Oper. Res. 21, 121–132 (1977)MATHGoogle Scholar
  7. 7.
    Burkard, R.E., Çela, E., Pardalos, P.M., Pitsoulis, L.S.: The Quadratic Assignment Problem. Springer, Heidelberg (1998)CrossRefMATHGoogle Scholar
  8. 8.
    Steinberg, L.: The backboard wiring problem: a placement algorithm. SIAM Rev. 3, 37–50 (1961)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Krarup, J., Pruzan, P.M.: Computer-aided layout design. In: Balinski, M.L., Lemarechal, C. (eds.) Mathematical Programming in Use. Mathematical Programming Studies, vol. 9, pp. 75–94. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  10. 10.
    Elshafei, A.N.: Hospital layout as a quadratic assignment problem. Oper. Res. Q. (1970–1977) 28, 167–179 (1977)CrossRefMATHGoogle Scholar
  11. 11.
    Burkard, R.E., Karisch, S.E., Rendl, F.: Qaplib - a quadratic assignment problemlibrary. J. Glob. Optim. 10, 391–403 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Anstreicher, K., Brixius, N., Goux, J.P., Linderoth, J.: Solving large quadratic assignment problems on computational grids. Math. Program. 91, 563–588 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Queyranne, M.: Performance ratio of polynomial heuristics for triangle inequality quadratic assignment problems. Oper. Res. Lett. 4, 231–234 (1986)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pardalos, P.M., Rendl, F., Wolkowicz, H.: The quadratic assignment problem: a survey and recent developments. In: Proceedings of the DIMACS Workshop on Quadratic Assignment Problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 1–42. American Mathematical Society (1994)Google Scholar
  15. 15.
    Commander, C.W.: A survey of the quadratic assignment problem, with applications. Morehead Electron. J. Appl. Math. 4, 1–15 (2005). MATH-2005-01Google Scholar
  16. 16.
    Loiola, E.M., de Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P., Querido, T.: A survey for the quadratic assignment problem. Eur. J. Oper. Res. 176, 657–690 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10, 305–313 (1962)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lawler, E.L.: The quadratic assignment problem. Manage. Sci. 9, 586–599 (1963)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, Y., Pardalos, P.M., Ramakrishnan, K.G., Resende, M.G.C.: Lower bounds for the quadratic assignment problem. Ann. Oper. Res. 50, 387–410 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Frieze, A., Yadegar, J.: On the quadratic assignment problem. Discrete Appl. Math. 5, 89–98 (1983)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kaufman, L., Broeckx, F.: An algorithm for the quadratic assignment problem using Benders’ decomposition. Eur. J. Oper. Res. 2, 204–211 (1978)CrossRefMATHGoogle Scholar
  22. 22.
    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2, 71–109 (1998)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 231–241 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Peng, J., Mittelmann, H., Li, X.: A new relaxation framework for quadratic assignment problems based on matrix splitting. Math. Program. Comput. 2, 59–77 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wolsey, L.A.: Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1998). A Wiley-Interscience PublicationMATHGoogle Scholar
  26. 26.
    ApS, M.: The MOSEK C optimizer API manual Version 7.1 (Revision 52) (2016)Google Scholar
  27. 27.
    Gurobi Optimization, I.: Gurobi optimizer reference manual (2016)Google Scholar
  28. 28.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307–335 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations