Advertisement

On Very Large Scale Assignment Problems

  • Yusin Lee
  • James B. Orlin

Abstract

In this paper we present computational testing results on very large scale random assignment problems. We consider a fully dense assignment problem with 2n nodes. Some conjectured or derived properties regarding fully dense assignment problems including the convergence of the optimal objective function value and the portion of nodes assigned with their kth best arc have been verified for networks up to n = 100,000 in size. Also we demonstrate the power of our approach in solving very large scale assignment problems by solving a one million node, one trillion arc random assignment problem.

Keywords

Assignment Problem Cost Distribution Short Path Tree Task Node Minimum Cost Flow Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. K. Ahuja, T. L. Magnanti, and J. B. Orlin (1993), Network Flows: Theory, Algorithms and Applications. Prentice Hall.MATHGoogle Scholar
  2. [2]
    M. Akgul . (1990), A Forest Primal-dual Algorithm for the Assignment Problem. Bilkent University, Ankara, Turkey, Research Report: IEOR-9014, 0 (0): 1–2.Google Scholar
  3. [3]
    F. Avram and D. J. Bertsimas (1993), On a Characterization of the Minimum Assignment and Matching in the Independent Random Model. In The third Symposium in Integer Programming and Combinatorial Optimization, Enrice, Italy.Google Scholar
  4. [4]
    M. L. Balinski (1985), “Signature Methods for the Assignment Problem,” Operations Research 33 (3) 527–536.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    D. P. Bertsekas (1990), “The Auction Algorithm for Assignment and other Network Flow Problems: A tutorial,” Interfaces 20 (4) 133–149.Google Scholar
  6. [6]
    D. J. Bertsimas 1993, Personal Communication.Google Scholar
  7. [7]
    H. N. Gabow and R. E. Tarjan (1989), “Faster Scaling Algorithms for Network Problems,” Siam Journal on Computing 18 (5) 1013–1036.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    M. S. Hung (1983), “A Polynomial Simplex Method for the Assignment Problem,” Operations Research 31 (3) 595–600.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    M. S. Hung and W. O. Rom (1980), “Solving the Assignment Problem by Relaxation,” Operations Research 28 (4) 969–982.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    R. M. Karp (1980), “An Algorithm to solve the Mxn Assignment Problem in Expected Time 0 (mn log n),” Networks 10 143–152.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    R. M. Karp (1984), An Upper Bound on the Expected Cost of an Optimal Assignment Technical Report, Computer Science Division, University of California, Berkeley.Google Scholar
  12. [12]
    J. Kennington and Z. Wang (1992), “A Shortest Augmenting Path Algorithm for the Semi-assignment Problem,” Operations Research 40 (1) 178–187.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Y. Lee and J. B. Orlin (1993), Quick Match: A Very Fast Algorithm for the Assignment Problem. Submitted to Mathematical Programming.Google Scholar
  14. [14]
    V. Lotfi (1989), “A Labeling Algorithm to solve the Assignment Problem,” Computers and Operations Research 16 (5) 397–408.MATHCrossRefGoogle Scholar
  15. [15]
    G. M. Megson and D. J. Evans (1990), “A Systolic Array Solution for the Assignment Problem,” The Computer Journal 33 (6) 562–569.MathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Mezard and G. Parisi (1985), “Replicas and Optimization,” Journal de Physique Lettres 771–778.Google Scholar
  17. [17]
    B. Olin (1992), Asymptotic Properties of Random Assignment Problems. PhD thesis, Royal Institute of Technology, Stockholm, Sweden.Google Scholar
  18. [18]
    P. M. Pardalos and K. G. Ramakrishnan (1993), On the Expected Optimal Value of Random Assignment Problems: Experimental Results and Open Questions.Google Scholar
  19. [19]
    K. G. Ramakrishnan, N. K. Karmarkar, and A. P. Kamath (1992), An Approximate Dual Projective Algorithm for solving Assignment Problems. Technical Report 92–4, DIMACS.Google Scholar
  20. [20]
    M. G. C. Resende and G. Veiga (1992), Computational Investigation of an Interior Point Linear Programming Algorithm for Minimum Cost Network Flows. Technical Report 92–4, Dimacs.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Yusin Lee
    • 1
  • James B. Orlin
    • 2
  1. 1.Department of Civil EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Sloan School of ManagementMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations