Skip to main content
SpringerLink
Log in

New approximation results on graph matching and related problems

  • Conference paper
  • First Online:
Book cover Graph-Theoretic Concepts in Computer Science (WG 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 903))

Included in the following conference series:

Abstract

For a graph G with e edges and n vertices, a maximum cardinality matching of G is a maximum subset M of edges such that no two edges of M are incident at a common vertex. The best known algorithm for solving the problem in general graphs requires O(n 5/2) time. We first propose an approximate maximum cardinality matching algorithm that runs in O(e+n) sequential time yielding a matching of size at least e/n−1, improving the bound known before. For bipartite graphs, the algorithm yields a matching of size at least 2e/n. The proposed algorithms are extremely simple, and the derived lowerbounds are existentially tight. Next, the proposed maximum cardinality matching algorithm is extended to the weighted case running in O(e+n) time. The problem of approximate maximum matching has a number of applications, for example in, Vertex Cover, TSP, MAXCUT, and VLSI physical design problems.

This work has been supported in part by the National Science Foundation under Grant MIP-9207267.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Avis. “Worst Case Bounds for the Euclidean Matching Problem”. In International J. Comput. Math. Appl., 7:251–257, 1981.

    Google Scholar 

  2. D. Avis. “A Survey of Heuristics for the Weighted Matching Problem”. In Networks, 13:475–493, 1983.

    Google Scholar 

  3. C. Berge. “Two Theorems in Graph Theory”. In Proc. Nat. Acad. Sci. 43, pages 842–844, 1957.

    Google Scholar 

  4. B. Bollobás. Extremal Graph Theory. London, NY; Academic Press, page 58, 1978.

    Google Scholar 

  5. J. D. Cho, S. Raje, and M. Sarrafzadeh. “A Generalized Multi-Way Maxcut Partitioning”. manuscript, EECS Dept. Northwestern Univ., April 1993.

    Google Scholar 

  6. N. Christfides. “Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem”. Technical Report, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA, 1976.

    Google Scholar 

  7. M. Fürer and B. Raghavachari. “Approximating the Minimum Degree Spanning Tree to within one from the Optimal Degree”. In Ann. ACM-SIAM Symp. on Discrete Algorithm, volume 4, pages 317–324, 1992.

    Google Scholar 

  8. H. N. Gabow. “Implementation of Algorithm for Maximum Matching on Ninbi-partite Graphs”. Ph.D. Thesis, Stanford University, 1973.

    Google Scholar 

  9. H. N. Gabow. “An Efficient Implementation of Edmonds' Algorithm for Maximum Matching on Graphs”. Journal of ACM, 23:221–234, 1976.

    Google Scholar 

  10. Z. Galil. “Efficient Algorithms for Finding Maximum Matching in Graphs”. Computing Surveys, 18:23–38, 1987.

    Google Scholar 

  11. M. K. Goldberg, and M. Burstein. “Heuristic Improvement Technique for Bisection of VLSI Networks”. In Proc. of the IEEE International Conference on Computer Design, 122–125, 1983.

    Google Scholar 

  12. H. N. Gabow, Z. Galil and T. H. Spencer. “Efficient Implementation of Graph Algorithms using Contraction”. In Proc. of the 25th Annual IEEE Symposium on Foundations of Computer Science, 347–357, 1984.

    Google Scholar 

  13. M. R. Garey, D. S. Johnson, and L. Stockmeyer. “Some Simplified NP-Complete Graph Problems”. Theoretical Computer Science, 1:237–267, 1976.

    Google Scholar 

  14. M. R. Garey, D. S. Johnson. “Computer and Intractability, A Guide to the Theory of NP-Completeness”. W. H. Freeman and Company, San Francisco, 1979.

    Google Scholar 

  15. D. Goldfard. “Efficient Dual Simplex Algorithms for the Assignment Problem”. Math. Program., 33:187–203, 1985.

    Google Scholar 

  16. M. X. Goemans and D. P. Williamson. “A General Approximation Technique for Constrained Forest Problems”. In Ann. ACM-SIAM Symp. on Discrete Algorithm, volume 4, pages 307–316, 1992.

    Google Scholar 

  17. D. J. Hagin and S. M. Venkatesan. “Approximation and Intractability Results for the Maximum Cut Problem and Its Variants”. IEEE Transactions on Computers, 40(1):110–113, January 1991.

    Google Scholar 

  18. M. M. Halldórsson. “A Still Better Performance Guarantee for Approximate Graph Coloring”. Information Processing Letters, 45:19–23, 1993.

    Google Scholar 

  19. J. E. Hopcroft and R. M. Karp. “N 5/2 algorithm for maximum matchings in bipartite graphs”. SIAM Journal on Computing, 2:225–231, 1973.

    Google Scholar 

  20. T. Lengauer. “Combinatorial Algorithms for Integrated Circuit Layout”. Applicable Theory in Computer Science, John Wiley & Sons, 273–274, 1990.

    Google Scholar 

  21. R. J. Lipton and R. E. Tarjan. “A Separator Theorem for Planar Graphs”. SIAM Journal on Applied Mathematics, 36(2): 177–189, April 1979.

    Google Scholar 

  22. R. J. Lipton and R. E. Tarjan. “Applications of a Planar Separator Theorem”. SIAM Journal on Computing, 9:615–627, 1980.

    Google Scholar 

  23. L. Lovasz. “Matching Structure and the Matching lattice”. Journal of Combinatorial Theory (b), 43:187–222, 1987.

    Google Scholar 

  24. L. Lovasz and M. D. Plummer. Matching Theory, chapter “Chapter 3: Size and Structure of Maximum Matchings”. Noth-Holland Mathematics Studies, Annals of Discrete Mathematics (29), 1986.

    Google Scholar 

  25. T. Mastumoto, N. Saigan, and K. Tsuji. “A New Efficient Approximation Algorithm for the Steiner Tree Problem in Rectilinear Graph”. In International Symposium on Circuits and Systems, pages 2874–2876. IEEE, 1990.

    Google Scholar 

  26. S. Micali and V. V. Vazirani. “An O(√∥V∥∥E∥) Algorithm for Finding Maximum Matching in General Graphs”. In Proceedings of 21th Annual Symposium on the Foundations of Computer Science, pages 17–27. IEEE, 1980.

    Google Scholar 

  27. T. Nishizeki. “Lower Bounds on the Cardinality of the Maximum Matchings of Graphs”. In Proceedings of 9th Southeastern Conference on Combinatorics, Graph Theory, and Computing, 1978.

    Google Scholar 

  28. S. M. Venkatesan. “Approximation Algorithms for Weighted Matching”. Theoretical Computer Science, 54:129–137, 1987.

    Google Scholar 

  29. P. M. B. Vitányi. “How Well Can A Graph Be n-Colored?”. Discrete Mathematics, 34:69–80, 1981.

    Google Scholar 

  30. M. Yannakakis. “On the Approximation of Maximum Satisfiability”. In Proc. 3rd Annual ACM-SIAM Symp. on Discrete Algorithms, pages 1–8, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Hokuriku, Japan

    Yoji Kajitani

  2. CAE, Samsung Electronics, Buchun Kyunggi-Do, Korea

    Jun Dong Cho

  3. Northwestern University, 60208, Evanston, IL

    Majid Sarrafzadeh

Authors
  1. Yoji Kajitani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Dong Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Majid Sarrafzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ernst W. Mayr Gunther Schmidt Gottfried Tinhofer

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kajitani, Y., Cho, J.D., Sarrafzadeh, M. (1995). New approximation results on graph matching and related problems. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 1994. Lecture Notes in Computer Science, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59071-4_60

Download citation

  • DOI: https://doi.org/10.1007/3-540-59071-4_60

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59071-2

  • Online ISBN: 978-3-540-49183-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation