Skip to main content
Log in

K best solutions to combinatorial optimization problems

  • Combinatorial Optimization
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

We review the Lawler-Murty [24,20] procedure for finding theK best solutions to combinatorial optimization problems. Then we introduce an alternative algorithm which is based on a binary search tree procedure. We apply both algorithms to the problems of finding theK best bases in a matroid, perfect matchings, and best cuts in a network.

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.

Similar content being viewed by others

References

  1. C. Berge,Graphes et Hypergraphes, 2nd Edition (Dunod, Paris, 1973).

    Google Scholar 

  2. R.E. Burkard and U. Zimmermann, Combinatorial optimization in linearly ordered semimodules: A survey, in:Modern Applied Mathematics: Optimization and Operations Research, ed. B. Korte (North-Holland, Amsterdam, 1982), p. 391.

    Google Scholar 

  3. P.M. Camerini, L. Fratta and F. Maffioli, TheK shortest spanning trees of a graph, Int. Rep. 73–10, IEE-LCE, Politeonico di Milano (1973).

  4. P.M. Camerini, L. Fratta and F. Maffioli, Efficient methods for ranking trees,Proc. 3rd Int. Symposium on Network Theory, Split, Yugoslavia (1975) p. 419.

  5. P.M. Camerini, L. Fratta and F. Maffioli, Ranking arborescences inO(K m logn) time,Eur. J. Oper. Res. 4(1980)235.

    Google Scholar 

  6. P.M. Camerini, L. Fratta and F. Maffioli, TheK best spanning arborescences of a network, Networks 10(1980)91.

    Google Scholar 

  7. P. Carraresi and C. Sodini, A binary enumeration tree to findK shortest paths, 7th Symposium on Operations Research, St. Gallen, Switzerland (1982).

  8. U. Derigs, Shortest augmenting paths and sensitivity analysis for optimal matchings, Report No. 82222-OR, Institut für OR, Universität Bonn (1982).

  9. J. Edmonds, Path, trees and flowers, Can. J. Math. 17(1965)449.

    Google Scholar 

  10. J. Edmonds, Matroids and the greedy algorithm, Math. Progr. 1(1971)127.

    Google Scholar 

  11. H.N. Gabow, Two algorithms for generating weighted spanning trees in order, SIAM J. Comput. 6(1977)139.

    Google Scholar 

  12. H. Hamacher, Flows in regular matroids, in:Math. Systems in Economics 69 (Oelgeschlager, Gunn and Hain, Cambridge, 1981).

    Google Scholar 

  13. H. Hamacher, J.C. Picard and M. Queyranne, On finding theK best cuts in a network, Oper. Res. Lett. 2(1984)303.

    Google Scholar 

  14. H. Hamacher, J.C. Picard and M. Queyranne, Ranking the cuts and cut-sets of a network, Ann. Discr. Appl. Math. 19(1984)183.

    Google Scholar 

  15. P.L. Hammer (Ivanescu), Some network flow problems solved with pseudo-Boolean programming, Oper. Res. 13(1965)388.

    Google Scholar 

  16. H. Ishii, A new method for finding theKth best paths in a graph, J. Oper. Res. Soc. Japan 21, 4(1978)469.

    Google Scholar 

  17. N. Katon, T. Ibaraki and H. Mine, AnO(K · n 2) algorithm forK shortest simple paths, Report, Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Japan (1978).

    Google Scholar 

  18. N. Katoh, T. Ibaraki and H. Mine, An algorithm for findingK minimum spanning trees, SIAM J. Comput. 10, 2(1981)247.

    Google Scholar 

  19. V. Klee, Combinatorial optimizations: What is the state of the art, MOR 5, 1(1980)1.

    Google Scholar 

  20. E.L. Lawler, A procedure for computing theK best solutions to discrete optimization problems and its application to the shortest path problem, Management Science 18(1972)401.

    Google Scholar 

  21. E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).

    Google Scholar 

  22. N. Megiddo, A. Tamir and R. Chandrasekaran, AnO(n log2 n) algorithm for theKth longest path in a tree with applications to location problems, SIAM J. Comput. 10, 2(1981)328.

    Google Scholar 

  23. K.G. Murty, Solving the fixed charge problem by ranking the extreme points, Oper. Res. 16, 2(1968)268.

    Google Scholar 

  24. K.G. Murty, An algorithm for ranking all the assignments in increasing order of cost, Oper. Res. 16(1968)682.

    Google Scholar 

  25. J.C. Picard and M. Queyranne, Selected applications of minimum cuts in networks, INFOR. 20, 4(1982)110.

    Google Scholar 

  26. D.D. Sleator and R.E. Tarjan, A data structure for dynamic trees, J. Computer and Systems Science, to appear; see alsoProc. 13th ACM Symposium on Theory of Computing (1981) p. 114.

  27. W.T. Tutte,Introduction to the Theory of Matroids (American Elsevier, New York, 1971).

    Google Scholar 

  28. G.M. Weber, Sensitivity analysis of optimal matching, Networks 11(1981)41.

    Google Scholar 

  29. D.J.A. Welsh,Matroid Theory (Academic Press, London, 1976).

    Google Scholar 

  30. U. Zimmermann, Linear and combinatorial optimization in ordered algebraic structures, Ann. Discr. Math. 10(1981).

  31. U. Derigs, Some basic exchange properties in combinatorial optimization and their application to constructing theK best solutions, Research Report No. 83295-OR, Institut für Oekonometrie und Operations Research, Universität Bonn (1983).

  32. D. Granot, A new characterization for matroids with applications, Research Report, Faculty of Commerce and Business Administration, The University of British Columbia (1982).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by the National Science Foundation, No. ECS-8412926.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hamacher, H.W., Queyranne, M. K best solutions to combinatorial optimization problems. Ann Oper Res 4, 123–143 (1985). https://doi.org/10.1007/BF02022039

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02022039

Keywords and phrases

Navigation