Skip to main content
Log in

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

Ein Algorithmus mit kürzesten alternierenden Wegen für dichte und dünne Zuordnungsprobleme

  • Contributed Papers
  • Published:
Computing Aims and scope Submit manuscript

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.

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balinski, M. L.: Signature methods for the assignment problem. Operations Research33, 527–536 (1985).

    Google Scholar 

  2. Barr, R., Glover, F., Klingman, D.: The alternating path basis algorithm for assignment problems. Mathematical Programming13, 1–13 (1977).

    Article  Google Scholar 

  3. Bertsekas, D. P.: A new algorithm for the linear assignment problem. Mathematical Programming21, 152–171 (1981).

    Article  Google Scholar 

  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. Carpaneto, G., Toth, P.: Algorithm 548 (solution of the assignment problem). ACM Transactions on Mathematical Software6, 104–111 (1980).

    Article  Google Scholar 

  6. Carpaneto, G., Toth, P.: Algorithm 50: algorithm for the solution of the assignment problem for sparse matrices. Computing31, 83–94 (1983).

    Article  Google Scholar 

  7. Carraresi, P., Sodini, C.: An efficient algorithm for the bipartite matching problem. European Journal of Operational Research23, 86–93 (1986).

    Article  Google Scholar 

  8. Derigs, U., Metz, A.: An efficient labeling technique for solving sparse assignment problems. Computing36, 301–311 (1986).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  10. Dorhout, B.: Het Lineaire Toewijzingsprobleem: Vergelijking van Algorithmen. Rapport BN 21/73, Mathematisch Centrum, Amsterdam (1973).

    Google Scholar 

  11. Dorhout, B.: Experiments with some algorithms for the linear assignment problem. Report BW 39, Mathematisch Centrum, Amsterdam (1975).

    Google Scholar 

  12. Dijkstra, E. W.: A note on two problems in connexion with graphs. Numerische Mathematik1, 269–271 (1959).

    Article  Google Scholar 

  13. Edmonds, J., Karp, R. M.: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM19, 248–264 (1972).

    Article  Google Scholar 

  14. Ford jr., L. R., Fulkerson, D. R.: Flows in Networks. Princeton: Princeton University Press 1962.

    Google Scholar 

  15. Glover, F., Klingman, D., Phillips, N.: A new polynomially bounded shortest path algorithm. Operations Research33, 65–73 (1985).

    Google Scholar 

  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. Goldfarb, D.: Efficient dual simplex algorithms for the assignment problem. Mathematical Programming33, 187–203 (1985).

    Article  Google Scholar 

  18. Hung, M. S., Rom, W. O.: Solving the assignment problem by relaxation. Operations Research28, 969–982 (1980).

    Google Scholar 

  19. Jonker, R.: Traveling salesman and assignment algorithms: design and implementation. Faculty of Actuarial Science and Econometrics, University of Amsterdam (1986).

  20. Jonker, R., Volgenant, A.: Improving the Hungarian assignment algorithm. Operations Research Letters5, 171–175 (1986).

    Article  Google Scholar 

  21. Karp, R. M.: An algorithm to solve them×n assignment problem in expected timeO (mn logn). Networks10, 143–152 (1980).

    Google Scholar 

  22. Kuhn, H. W.: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly2, 83–97 (1955).

    Google Scholar 

  23. Lawler, E. L.: Combinatorial Optimization: Networks and Matroids. New York: Holt, Rinehart & Winston 1976.

    Google Scholar 

  24. Mack, C.: The Bradford method for the assignment problem. New Journal of Statistics and Operational Research1, 17–29 (1969).

    Google Scholar 

  25. McGinnis, L. F.: Implementation and testing of a primal-dual algorithm for the assignment problem. Operations Research31, 277–291 (1983).

    Google Scholar 

  26. Papadimitriou, C. H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, N. J.: Prentice-Hall 1982.

    Google Scholar 

  27. Silver, R.: An algorithm for the assignment problem. Communications of the ACM3, 605–606 (1960).

    Article  Google Scholar 

  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. Tomizawa, N.: On some techniques useful for the solution of transportation problems. Networks1, 173–194 (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jonker, R., Volgenant, A. A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38, 325–340 (1987). https://doi.org/10.1007/BF02278710

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02278710

AMS Subject Classifications

Key words

Navigation