Skip to main content
Springer Nature Link
Log in

Efficient algorithms for graphic matroid intersection and parity

  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1985)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 194))

Included in the following conference series:

Abstract

An algorithm for matroid intersection, based on the phase approach of Dinic for network flow and Hopcroft and Karp for matching, is presented. An implementation for graphic matroids uses time O(n 1/2 m) if m is Ω(n 3/2 lg n), and similar expressions otherwise. An implementation to find k edge-disjoint spanning trees on a graph uses time O(k 3/2 n 1/2 m) if m is Ω(n lg n) and a similar expression otherwise; when m is O(k 1/2 n 3/2) this improves the previous bound, O(k 2 n 2). Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network. An algorithm for graphic matroid parity is presented that runs in time O(n m lg 5 n). This improves the previous bound of O(n 2 m).

This work was supported in part by NSF Grant #MCS-8302648.

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. M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979.

    Google Scholar 

  2. J. Bruno and L. Weinberg, “A constructive graph-theoretic solution of the Shannon switching game”, IEEE Trans. on Circuit Theory CT-17, 1, 1970, pp.74–81.

    Google Scholar 

  3. W.H.Cunningham, “Matroid partition and intersection algorithms”, Carleton University, preprint.

    Google Scholar 

  4. E.A. Dinic, “Algorithm for solution of a problem of maximum flow in a network with power estimation”, Sov. Math. Dokl. 11, 5, 1970, pp.1277–1280.

    Google Scholar 

  5. S. Even and R.E. Tarjan, “Network flow and testing graph connectivity”, SIAM J. Comput. 4, 1975, pp.507–518.

    Article  Google Scholar 

  6. G.N.Frederickson, “Data structures for on-line updating of minimum spanning trees”, Proc. 15 th Annual Symp. on Theory of Computing, 1983, pp.252–257.

    Google Scholar 

  7. H.N.Gabow and R.E.Tarjan, “A linear-time algorithm for a special case of disjoint set union”, Proc. 15 th Annual ACM Symp. on Theory of Computing, 1983, pp.246–251.

    Google Scholar 

  8. H.N. Gabow and R.E. Tarjan, “Efficient algorithms for a family of matroid intersection problems”, J.Algorithms 5, 1984, pp. 80–131.

    Article  Google Scholar 

  9. J. Hopcroft and R. Karp, “An n 5/2 algorithm for maximum matchings in bipartite graphs”, SIAM J. Comput.2, 1973, pp.225–231.

    Article  Google Scholar 

  10. H. Imai, “Network-flow algorithms for lower-truncated transversal polymatroids”, J. Operations Res. Soc. Japan 26, 3,1983, pp.186–210.

    Google Scholar 

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

    Google Scholar 

  12. L.Lovasz, “The matroid matching problem”, in Algebraic Methods in Graph Theory, Colloquia Mathematica Societatis Janos Bolyai, Szeged, Hungary, 1978, pp.495–517.

    Google Scholar 

  13. G.S.Lueker, “A data structure for orthogonal range queries”, Proc. 19 th Annual Symp. on Foundations of Comp. Sci., 1978, pp.28–34.

    Google Scholar 

  14. E.L. Lawler and C.U. Martel, “Computing maximal ‘polymatroidal’ network flows”, Math. of Operations Research 7, 1982, pp.334–347.

    Google Scholar 

  15. J.Roskind and R.E.Tarjan, “A note on finding minimum-cost edge-disjoint spanning trees”, Math. of Operations Research, to appear.

    Google Scholar 

  16. M.Stallmann and H.N.Gabow, “An augmenting path algorithm for the parity problem on linear matroids”, Proc. 25 th Annual Symp. on Foundations of Comp. Sci., 1984, pp.217–228.

    Google Scholar 

  17. D.D. Sleator and R.E. Tarjan, “A data structure for dynamic trees”, J. Comp. and System Sci. 26, 1983, pp.362–391.

    Article  Google Scholar 

  18. J.W. Suurballe and R.E. Tarjan, “A quick method for finding shortest pairs of paths”, Networks 14, 1984, pp.325–336.

    Google Scholar 

  19. R.E.Tarjan, Data Structures and Network Algorithms, SIAM Monograph, Philadelphia, Pa., 1983.

    Google Scholar 

  20. C.A.Tovey and M.A.Trick, “An O(m 4 d) algorithm for the maximum polymatroidal flow problem”, Georgia Institute of Technology, preprint.

    Google Scholar 

  21. D.J.A. Welsh, Matroid Theory, Academic Press, New York, 1976.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Colorado, 80309, Boulder, CO, USA

    Harold N. Gabow

  2. Department of Computer Science, North Carolina State University, 27695-8206, Raleigh, NC, USA

    Matthias Stallmann

Authors
  1. Harold N. Gabow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias Stallmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Wilfried Brauer

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gabow, H.N., Stallmann, M. (1985). Efficient algorithms for graphic matroid intersection and parity. In: Brauer, W. (eds) Automata, Languages and Programming. ICALP 1985. Lecture Notes in Computer Science, vol 194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015746

Download citation

  • DOI: https://doi.org/10.1007/BFb0015746

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15650-5

  • Online ISBN: 978-3-540-39557-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation