Computing

, Volume 38, Issue 4, pp 325–340 | Cite as

A shortest augmenting path algorithm for dense and sparse linear assignment problems

  • R. Jonker
  • A. Volgenant
Contributed Papers

Abstract

We develop a shortest augmenting path algorithm for the linear assignment problem. It contains new initialization routines and a special implementation of Dijkstra's shortest path method. For both dense and sparse problems computational experiments show this algorithm to be uniformly faster than the best algorithms from the literature. A Pascal implementation is presented.

AMS Subject Classifications

90 C 08 68 E 10 

Key words

Linear assignment problem shortest path methods Pascal implementation 

Ein Algorithmus mit kürzesten alternierenden Wegen für dichte und dünne Zuordnungsprobleme

Zusammenfassung

Wir entwickeln einen Algorithmus mit kürzesten alternierenden Wegen für das lineare Zuordnungsproblem. Er enthält neue Routinen für die Anfangswerte und eine spezielle Implementierung der Kürzesten-Wege-Methode von Dijkstra. Sowohl für dichte als auch für dünne Probleme zeigen Testläufe, daß unser Algorithmus gleichmäßig schneller als die besten Algorithmen aus der Literatur ist. Eine Implementierung in Pascal wird angegeben.

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References

  1. [1]
    Balinski, M. L.: Signature methods for the assignment problem. Operations Research33, 527–536 (1985).Google Scholar
  2. [2]
    Barr, R., Glover, F., Klingman, D.: The alternating path basis algorithm for assignment problems. Mathematical Programming13, 1–13 (1977).CrossRefGoogle Scholar
  3. [3]
    Bertsekas, D. P.: A new algorithm for the linear assignment problem. Mathematical Programming21, 152–171 (1981).CrossRefGoogle Scholar
  4. [4]
    Burkard, R. E., Derigs, U.: Assignment and Matching Problems: Solution Methods with Fortran Programs, pp. 1–11. Berlin-Heidelberg-New York: Springer 1980.Google Scholar
  5. [5]
    Carpaneto, G., Toth, P.: Algorithm 548 (solution of the assignment problem). ACM Transactions on Mathematical Software6, 104–111 (1980).CrossRefGoogle Scholar
  6. [6]
    Carpaneto, G., Toth, P.: Algorithm 50: algorithm for the solution of the assignment problem for sparse matrices. Computing31, 83–94 (1983).CrossRefGoogle Scholar
  7. [7]
    Carraresi, P., Sodini, C.: An efficient algorithm for the bipartite matching problem. European Journal of Operational Research23, 86–93 (1986).CrossRefGoogle Scholar
  8. [8]
    Derigs, U., Metz, A.: An efficient labeling technique for solving sparse assignment problems. Computing36, 301–311 (1986).CrossRefGoogle Scholar
  9. [9]
    Derigs, U., Metz, A.: An in-core/out-of-core method for solving large scale assignment problems. Zeitschrift für Operations Research30, 181–195 (1986).CrossRefGoogle Scholar
  10. [10]
    Dorhout, B.: Het Lineaire Toewijzingsprobleem: Vergelijking van Algorithmen. Rapport BN 21/73, Mathematisch Centrum, Amsterdam (1973).Google Scholar
  11. [11]
    Dorhout, B.: Experiments with some algorithms for the linear assignment problem. Report BW 39, Mathematisch Centrum, Amsterdam (1975).Google Scholar
  12. [12]
    Dijkstra, E. W.: A note on two problems in connexion with graphs. Numerische Mathematik1, 269–271 (1959).CrossRefGoogle Scholar
  13. [13]
    Edmonds, J., Karp, R. M.: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM19, 248–264 (1972).CrossRefGoogle Scholar
  14. [14]
    Ford jr., L. R., Fulkerson, D. R.: Flows in Networks. Princeton: Princeton University Press 1962.Google Scholar
  15. [15]
    Glover, F., Klingman, D., Phillips, N.: A new polynomially bounded shortest path algorithm. Operations Research33, 65–73 (1985).Google Scholar
  16. [16]
    Glover, F., Klingman, D., Phillips, N., Schneider, R.: New polynomial shortest path algorithms and their computational attributes. Management Science31, 1106–1128 (1985).Google Scholar
  17. [17]
    Goldfarb, D.: Efficient dual simplex algorithms for the assignment problem. Mathematical Programming33, 187–203 (1985).CrossRefGoogle Scholar
  18. [18]
    Hung, M. S., Rom, W. O.: Solving the assignment problem by relaxation. Operations Research28, 969–982 (1980).Google Scholar
  19. [19]
    Jonker, R.: Traveling salesman and assignment algorithms: design and implementation. Faculty of Actuarial Science and Econometrics, University of Amsterdam (1986).Google Scholar
  20. [20]
    Jonker, R., Volgenant, A.: Improving the Hungarian assignment algorithm. Operations Research Letters5, 171–175 (1986).CrossRefGoogle Scholar
  21. [21]
    Karp, R. M.: An algorithm to solve them×n assignment problem in expected timeO (mn logn). Networks10, 143–152 (1980).Google Scholar
  22. [22]
    Kuhn, H. W.: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly2, 83–97 (1955).Google Scholar
  23. [23]
    Lawler, E. L.: Combinatorial Optimization: Networks and Matroids. New York: Holt, Rinehart & Winston 1976.Google Scholar
  24. [24]
    Mack, C.: The Bradford method for the assignment problem. New Journal of Statistics and Operational Research1, 17–29 (1969).Google Scholar
  25. [25]
    McGinnis, L. F.: Implementation and testing of a primal-dual algorithm for the assignment problem. Operations Research31, 277–291 (1983).Google Scholar
  26. [26]
    Papadimitriou, C. H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, N. J.: Prentice-Hall 1982.Google Scholar
  27. [27]
    Silver, R.: An algorithm for the assignment problem. Communications of the ACM3, 605–606 (1960).CrossRefGoogle Scholar
  28. [28]
    Tabourier, Y.: Un Algorithme pour le Problème d'Affectation. R.A.I.R.O. Recherche Opérationnelle/Operations Research6, 3–15 (1972).Google Scholar
  29. [29]
    Tomizawa, N.: On some techniques useful for the solution of transportation problems. Networks1, 173–194 (1971).Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • R. Jonker
    • 1
  • A. Volgenant
    • 1
  1. 1.Faculty of Actuarial Sciences and EconometricsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Koninklyke/Shell-LaboratoriumAmsterdamThe Netherlands

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