Advertisement

Dynamic Sparsification for Quadratic Assignment Problems

  • Maximilian JohnEmail author
  • Andreas Karrenbauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

We present a framework for optimizing sparse quadratic assignment problems. We propose an iterative algorithm that dynamically generates the quadratic part of the assignment problem and, thus, solves a sparsified linearization of the original problem in every iteration. This procedure results in a hierarchy of lower bounds and, in addition, provides heuristic primal solutions in every iteration. This framework was motivated by the task of the French government to design the French keyboard standard, which included solving sparse quadratic assignment problems with over 100 special characters; a size where many commonly used approaches fail. The design of a new standard often involves conflicting opinions of multiple stakeholders in a committee. Hence, there is no agreement on a single well-defined objective function that can be used for an extensive one-shot optimization. Instead, the process is highly interactive and demands rapid prototyping, e.g., quick primal solutions, on-the-fly evaluation of manual changes, and prompt assessments of solution quality. Particularly concerning the latter aspect, our algorithm is able to provide high-quality lower bounds for these problems in several minutes.

Keywords

Quadratic assignment Integer programming Linearization Keyboard optimization 

References

  1. 1.
    Adams, W., Johnson, T.: Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS 512 Ser. Discret. Math. Theor. Comput. Sci. 16, 43–77 (1994). https://doi.org/10.1090/dimacs/016/02Google Scholar
  2. 2.
    AFNOR: Interfaces utilisateurs - Dispositions de clavier bureautique français, NF Z71–300 Avril 2019Google Scholar
  3. 3.
    Arkin, E.M., Hassin, R., Sviridenko, M.: Approximating the maximum quadratic assignment problem. Inf. Process. Lett. 77(1), 13–16 (2001).  https://doi.org/10.1016/S0020-0190(00)00151-4MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Birkhoff, D.: Tres observaciones sobre el algebra lineal. Universidad Nacional de Tucuman Revista Serie A 5, 147–151 (1946)Google Scholar
  5. 5.
    Burkard, R.E., Çela, E., Pardalos, P.M., Pitsoulis, L.S.: The Quadratic Assignment Problem, pp. 1713–1809. Springer, Boston (1998). https://doi.org/10.1007/978-1-4613-0303-9_27CrossRefGoogle Scholar
  6. 6.
    Burkard, R., Offermann, J.: Entwurf von Schreibmaschinentastaturen mittels quadratischer Zuordnungsprobleme. Zeitschrift für Oper. Res. 21, 121–132 (1977)zbMATHGoogle Scholar
  7. 7.
    DGLFLF: Rapport au Parlement sur l’emploi de la langue française. Government Report (2015). http://www.culture.gouv.fr/Thematiques/Langue-francaise-et-langues-de-France/La-DGLFLF/Nos-priorites/Rapport-au-Parlement-sur-l-emploi-de-la-langue-francaise-2015. From the Délégation générale à la langue française et aux langues de France of the Ministère de la Culture et de la Communication (in French)
  8. 8.
    DGLFLF: Vers une norme française pour les claviers informatiques. Government Publication (2016). http://www.culture.gouv.fr/Thematiques/Langue-francaise-et-langues-de-France/Politiques-de-la-langue/Langues-et-numerique/Les-technologies-de-la-langue-et-la-normalisation/Vers-une-norme-francaise-pour-les-claviers-informatiques. From the Délégation générale à la langue française et aux langues de France of the Ministère de la Culture et de la Communication (in French)
  9. 9.
    Feit, A.M.: Assignment Problems for Optimizing Text Input. G5 artikkeliväitöskirja (2018). http://urn.fi/URN:ISBN:978-952-60-8016-1
  10. 10.
    Frieze, A., Yadegar, J.: On the quadratic assignment problem. Discrete Appl. Math. 5(1), 89–98 (1983).  https://doi.org/10.1016/0166-218X(83)90018-5MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10, 305–313 (1962)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gurobi Optimization, L.: Gurobi Optimizer Version 8.1 (2019). http://www.gurobi.com
  13. 13.
    Huber, C., Riedl, W.: The Quadratic Assignment Problem: the Linearization of Xia and Yuan is Weaker than the Linearization of Adams and Johnson and a Family of Cuts to Narrow the Gap, preprint on webpage at https://arxiv.org/abs/1710.02472
  14. 14.
    John, M., Karrenbauer, A.: A Novel SDP Relaxation for the Quadratic Assignment Problem Using Cut Pseudo Bases, pp. 414–425. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45587-7_36Google Scholar
  15. 15.
    Kaufman, L., Broeckx, F.: An algorithm for the quadratic assignment problem using Benders’ decomposition. Eur. J. Oper. Res. 2(3), 207–211 (1978).  https://doi.org/10.1016/0377-2217(78)90095-4CrossRefzbMATHGoogle Scholar
  16. 16.
    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). http://EconPapers.repec.org/RePEc:cwl:cwldpp:4
  17. 17.
    Lawler, E.L.: The quadratic assignment problem. Manag. Sci. 9(4), 586–599 (1963).  https://doi.org/10.1287/mnsc.9.4.586MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lee, Y., Orlin, J.B.: On Very Large Scale Assignment Problems, pp. 206–244. Springer, Boston (1994). https://doi.org/10.1007/978-1-4613-3632-7_12CrossRefGoogle Scholar
  19. 19.
    Nugent, C., Vollman, T., Ruml, J.: An experimental comparison of techniques for the assignment of facilities to locations. Oper. Res. 16(1), 150–173 (1968).  https://doi.org/10.1287/opre.16.1.150CrossRefGoogle Scholar
  20. 20.
    Peng, J., Mittelmann, H., Li, X.: A new relaxation framework for quadratic assignment problems based on matrix splitting. Math. Program. Comput. 2(1), 59–77 (2010).  https://doi.org/10.1007/s12532-010-0012-6MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pollatschek, M., Gershoni, N., Radday, Y.: Optimization of the typewriter keyboard by simulation. Angewandte Mathematik 10 (1976)Google Scholar
  22. 22.
    Povh, J., Rendl, F.: Copositive and Semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6(3), 231–241 (2009).  https://doi.org/10.1016/j.disopt.2009.01.002MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Queyranne, M.: Performance ratio of polynomial heuristics for triangle inequality quadratic assignment problems. Oper. Res. Lett. 4(5), 231–234 (1986).  https://doi.org/10.1016/0167-6377(86)90007-6MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems. Discret. Appl. Math. 52(1), 83–106 (1994).  https://doi.org/10.1016/0166-218X(92)00190-WMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xia, Y., Yuan, Y.X.: A new linearization method for quadratic assignment problems. Optim. Methods Softw. 21(5), 805–818 (2006).  https://doi.org/10.1080/10556780500273077MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, H., Beltran-Royo, C., Ma, L.: Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers. Ann. OR 207, 261–278 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2(1), 71–109 (1998).  https://doi.org/10.1023/A:1009795911987MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations