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A Cutting Plane Algorithm for the Linear Ordering Problem

Author

Listed:
  • Martin Grötschel

    (Universitat Augsburg, Augsburg, West Germany)

  • Michael Jünger

    (Universitat Augsburg, Augsburg, West Germany)

  • Gerhard Reinelt

    (Universitat Augsburg, Augsburg, West Germany)

Abstract

The linear ordering problem is an NP-hard combinatorial optimization problem with a large number of applications (including triangulation of input-output matrices, archaeological senation, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences). In a former paper, we have investigated the facet structure of the 0/1-polytope associated with the linear ordering problem. Here we report on a new algorithm that is based on these theoretical results. The main part of the algorithm is a cutting plane procedure using facet defining inequalities. This procedure is combined with various heuristics and branch and bound techniques. Our computational results compare favorably with the results of existing codes. In particular, we could triangulate all input-output matrices, of size up to 60 × 60, available to us within acceptable time bounds.

Suggested Citation

  • Martin Grötschel & Michael Jünger & Gerhard Reinelt, 1984. "A Cutting Plane Algorithm for the Linear Ordering Problem," Operations Research, INFORMS, vol. 32(6), pages 1195-1220, December.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:6:p:1195-1220
    DOI: 10.1287/opre.32.6.1195
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    Citations

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    Cited by:

    1. Javier Alcaraz & Eva M. García-Nové & Mercedes Landete & Juan F. Monge, 2020. "The linear ordering problem with clusters: a new partial ranking," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 646-671, October.
    2. Manfred Padberg, 2005. "Classical Cuts for Mixed-Integer Programming and Branch-and-Cut," Annals of Operations Research, Springer, vol. 139(1), pages 321-352, October.
    3. Miles William W & Fowks Gary T & Coulter Lisa O, 2010. "AccuV College Football Ranking Model," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 6(3), pages 1-17, July.
    4. Michael J. Brusco & Douglas Steinley & Ashley L. Watts, 2022. "Disentangling relationships in symptom networks using matrix permutation methods," Psychometrika, Springer;The Psychometric Society, vol. 87(1), pages 133-155, March.
    5. Aardal, K.I. & van Hoesel, S., 1995. "Polyhedral Techniques in Combinatorial Optimization," Other publications TiSEM ed028a07-eb6a-4c8d-8f21-d, Tilburg University, School of Economics and Management.
    6. Irène Charon & Olivier Hudry, 2010. "An updated survey on the linear ordering problem for weighted or unweighted tournaments," Annals of Operations Research, Springer, vol. 175(1), pages 107-158, March.
    7. Miguel F. Anjos & Anthony Vannelli, 2008. "Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 611-617, November.
    8. Jose Apesteguia & Miguel A. Ballester, 2015. "A Measure of Rationality and Welfare," Journal of Political Economy, University of Chicago Press, vol. 123(6), pages 1278-1310.
    9. George Steiner & Dan Wang & Yufei Yuan, 1992. "Primal dual algorithms for the vehicle refueling problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(7), pages 905-918, December.
    10. Abraham Duarte & Manuel Laguna & Rafael Martí, 2011. "Tabu search for the linear ordering problem with cumulative costs," Computational Optimization and Applications, Springer, vol. 48(3), pages 697-715, April.
    11. Fred Glover & Vicente Campos & Rafael Martí, 2021. "Tabu search tutorial. A Graph Drawing Application," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(2), pages 319-350, July.
    12. B. Jay Coleman, 2005. "Minimizing Game Score Violations in College Football Rankings," Interfaces, INFORMS, vol. 35(6), pages 483-496, December.
    13. Rajeev Kohli & Khaled Boughanmi & Vikram Kohli, 2019. "Randomized Algorithms for Lexicographic Inference," Operations Research, INFORMS, vol. 67(2), pages 357-375, March.
    14. Fabrizio Borghini & Isabel Méndez-Díaz & Paula Zabala, 2020. "An exact algorithm for the edge coloring by total labeling problem," Annals of Operations Research, Springer, vol. 286(1), pages 11-31, March.
    15. Elena Fernández & Justo Puerto & Antonio Rodríguez-Chía, 2013. "On discrete optimization with ordering," Annals of Operations Research, Springer, vol. 207(1), pages 83-96, August.
    16. Akbari, Sina & Escobedo, Adolfo R., 2023. "Beyond kemeny rank aggregation: A parameterizable-penalty framework for robust ranking aggregation with ties," Omega, Elsevier, vol. 119(C).
    17. P. Detti & D. Pacciarelli, 2001. "A branch and bound algorithm for the minimum storage‐time sequencing problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(4), pages 313-331, June.
    18. Vicente Campos & Manuel Laguna & Rafael Martí, 2005. "Context-Independent Scatter and Tabu Search for Permutation Problems," INFORMS Journal on Computing, INFORMS, vol. 17(1), pages 111-122, February.
    19. W. Art Chaovalitwongse & Carlos A. S. Oliveira & Bruno Chiarini & Panos M. Pardalos & Mauricio G. C. Resende, 2011. "Revised GRASP with path-relinking for the linear ordering problem," Journal of Combinatorial Optimization, Springer, vol. 22(4), pages 572-593, November.
    20. Ellis L. Johnson & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 2-23, February.
    21. Aardal, K. & van Hoesel, C.P.M., 1995. "Polyhedral techniques in combinatorial optimization," Research Memorandum 014, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).

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