The random linear bottleneck assignment problem

  • Ulrich Pferschy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)

Abstract

In this contribution asymptotic properties of the linear bottleneck assignment problem LBAP are investigated. It is shown that the expected value of the optimal solution of an n × n LBAP with independently and identically distributed costs tends towards the infimum of the cost range as n tends to infinity. Furthermore, explicit upper and lower bounds for the uniform cost distribution are given as functions in n. Exploiting results from evolutionary random graph theory an algorithm with O(n2) expected running time is presented.

Keywords

Random Bottleneck Assignment Average Case Analysis Random Graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical functions, Dover Publications, New York, 1965.Google Scholar
  2. 2.
    B. Bollobás, Random Graphs, Academic Press, 1985.Google Scholar
  3. 3.
    B. Bollobás, A. Thomason, Random graphs of small order. Random Graphs '83, Annals of Discrete Mathematics 28, 47–97, 1985.Google Scholar
  4. 4.
    U. Derigs, Alternate strategies for solving bottleneck assignment problems-analysis and computational results. Computing 33, 95–106, 1984.Google Scholar
  5. 5.
    J.B.G. Frenk, M. van Houweninge, A.H.G. Rinnooy Kan, Order statistics and the linear assignment problem. Report 8609/A, Econometric Institute, Erasmus University, Rotterdam, The Netherlands, 1986.Google Scholar
  6. 6.
    H.N. Gabow, R.E. Tarjan, Algorithms for two bottleneck optimization problems. J. of Algorithms 9, 411–417, 1988.Google Scholar
  7. 7.
    R.M. Karp, An algorithm to solve the m × n assignment problem in expected time O(mn log n). Networks 10, 143–152, 1980.Google Scholar
  8. 8.
    R.M. Karp, An upper bound on the expected cost of an optimal assignment. Technical report, Computer Science Division, Univ. of California, Berkeley, 1984.Google Scholar
  9. 9.
    A.J. Lazarus, The assignment problem with uniform (0, 1) cost matrix. Master's thesis, Department of Mathematics, Princeton University, 1979.Google Scholar
  10. 10.
    B. Olin, Asymptotic properties of random assignment problems. PhD-thesis, Division of Optimization and Systems Theory, Department of Mathematics, Royal Institute of Technology, Stockholm, 1992.Google Scholar
  11. 11.
    E.M. Palmer, Graphical Evolution, Wiley, 1985.Google Scholar
  12. 12.
    D.W. Walkup, On the expected value of a random assignment problem. SIAM J. Computing 8, 440–442, 1979.Google Scholar
  13. 13.
    D.W. Walkup, Matchings in random regular bipartite digraphs. Discrete Mathematics 31, 59–64, 1980.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Ulrich Pferschy
    • 1
  1. 1.Institute of Mathematics BTU GrazGrazAustria

Personalised recommendations