Mathematical Programming

, Volume 33, Issue 3, pp 243–259 | Cite as

Solving matching problems with linear programming

  • M. Grötschel
  • O. Holland
Article

Abstract

In this paper we describe an implementation of a cutting plane algorithm for the perfect matching problem which is based on the simplex method. The algorithm has the following features:
  • -It works on very sparse subgraphs ofKn which are determined heuristically, global optimality is checked using the reduced cost criterion.

  • -Cutting plane recognition is usually accomplished by heuristics. Only if these fail, the Padberg-Rao procedure is invoked to guarantee finite convergence.

Our computational study shows that—on the average—very few variables and very few cutting planes suffice to find a globally optimal solution. We could solve this way matching problems on complete graphs with up to 1000 nodes. Moreover, it turned out that our cutting plane algorithm is competitive with the fast combinatorial matching algorithms known to date.

Key words

Matching Linear Programming Cutting Plane Algorithms Computational Study 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ball and U. Derigs, “An analysis of alternate strategies for implementing matching algorithms”,Networks 13 (1983) 517–549.Google Scholar
  2. R.E. Burkard and U. Derigs,Assignment and Matching Problems: Solution Methods with Fortran-Programs (Springer Lecture Notes in Economics and Mathematical Systems, No. 184, Berlin, 1980).Google Scholar
  3. H. Crowder and M.W. Padberg, “Solving large-scale symmetric travelling salesman problems”,Management Science 26 (1980) 495–509.Google Scholar
  4. W. Cunningham and A. Marsh, “A primal algorithm for optimum matching”,Mathematical Programming Study 8 (1978) 50–72.Google Scholar
  5. U. Derigs, “Solving matching problems via shortest path techniques”, Report No. 83263-OR, Institut für Ökonometrie und Operations Research, Universität Bonn (Bonn, 1983).Google Scholar
  6. U. Derigs, “Solving large scale matching problems efficiently—A new primal matching approach”, Report No. 84346-OR, Institut für Ökonometrie und Operations Research, Universität Bonn (Bonn, 1984).Google Scholar
  7. E.A. Dinic, “Algorithm for solution of a problem of maximum flow in a network with power estimation”,Soviet Mathematics Doklady 11 (1970) 1277–1280.Google Scholar
  8. J. Edmonds, “Paths, trees and flowers”,Canadian Journal of Mathematics 17 (1965) 449–467.Google Scholar
  9. J. Edmonds, “Maximum matching and a polyhedron with 0, 1 vertices”,Journal of Research National Bureau of Standards 69B (1965) 125–130.Google Scholar
  10. L.R. Ford and D.R. Fulkerson, “A simple algorithm for finding maximal flows and an application to the Hitchcock problem”,Canadian Journal of Mathematics 9 (1957) 210–218.Google Scholar
  11. F. Glover, D. Klingman, J. Mote and D. Whitman, “A primal simplex variant for the maximum flow problem”, Center for Cybernetic Studies, CCS 362 (Austin, TX, 1979).Google Scholar
  12. M. Grötschel, M. Jünger and G. Reinelt, “A cutting plane algorithm for the linear ordering problem”,Operations Research 32 (1984) 1195–1220.Google Scholar
  13. M. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization”,Combinatorica 1 (1981) 169–197.Google Scholar
  14. M. Grötschel and W.R. Pulleyblank, “Weakly bipartite graphs and the max-cut problem”,Operations Research Letters 1 (1981) 23–27.Google Scholar
  15. A.V. Karzanov, “Determining the maximal flow in a network by the method of preflows”,Soviet Mathematics Doklady 15 (1972) 434–437.Google Scholar
  16. E.L. Lawler,Combinatorial optimization: Networks and matroids (Holt, Rinehart and Winston, New York, 1976).Google Scholar
  17. V.M. Malhorta, M.P. Kumar and S.N. Maheshwari, “An O(|V|3) algorithm for finding maximum flows in networks”,Information Processing Letters 7 (1978) 277–278.Google Scholar
  18. M.W. Padberg and M.R. Rao, “Odd minimum cut-sets andb-matchings”,Mathematics of Operations Research 7 (1982) 67–80.Google Scholar
  19. M.W. Padberg and M.R. Rao, “The Russian method for linear inequalities III: Bounded integer Programming”, Preprint, GBA New York University, New York, May 1981.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1985

Authors and Affiliations

  • M. Grötschel
    • 1
  • O. Holland
    • 2
  1. 1.Lehstuhl für Angewandte Mathematik IIUniversität AugsburgAugsburgFR Germany
  2. 2.Institut für Ökonometrie und Operations ResearchUniversität BonnBonnFR Germany

Personalised recommendations