Computing Girth and Cogirth in Perturbed Graphic Matroids

Abstract

We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    F. Barahona and M. Conforti: A construction for binary matroids, Discrete Math. 66 (1987), 213–218.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    M. Conforti and M. R. Rao: Some new matroids on graphs: Cut sets and the max cut problem, Math. Oper. Res. 12 (1987), 193–204.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. Geelen, B. Gerards and G. Whittle: The highly-connected matroids in minorclosed classes, Ann. Comb. 19 (2015), 107–123.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    C. D. Godsil: Algebraic Combinatorics, Chapman and Hall, New York, 1993.

    MATH  Google Scholar 

  5. [5]

    D. R. Karger: Global min-cuts in RNC, and other ramifications of a simple mincut algorithm, in: Proc. 4th Annu. ACM-SIAM Symposium on Discrete Algorithms, ACM-SIAM, 1993, 84–93.

    Google Scholar 

  6. [6]

    D. R. Karger and C. Stein: A new approach to the minimum cut problem, J. ACM 43 (1996), 601–640.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    L. Lovász: On determinants, matchings, and random algorithms, in: Fundamentals of Computation Theory (L. Budach, ed.), Akademie-Verlag, Berlin, 1979, 565–574.

    Google Scholar 

  8. [8]

    K. Mulmuley, U. V. Vazirani and V. V. Vazirani: Matching is as easy as matrix inversion, in: Proc. STOC’ 87 Proceedings of the nineteenth annual ACM symposium on Theory of computing, ACM, New York, 1987, 345–354.

    Google Scholar 

  9. [9]

    M. Mahajan, P. R. Subramanya and V. Vinay: The combinatorial approach yields an NC algorithm for computing Pfaffians, Discrete Appl. Math. 143 (2004), 1–16.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    M. W. Padberg and M. R. Rao: Odd minimum cut-sets and b-matchings, Math. Oper. Res. 1 (1982), 67–80.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    J. T. Schwartz: Fast probabilistic algorithms for verification of polynomial identities, J. ACM 27 (1980), 701–717.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    W. T. Tutte: The factorization of linear graphs, J. London Math. Soc. 21 (1947), 107–111.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    A. Vardy: The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43 (1997), 1757–1766.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    R. Zippel: Probabilistic algorithms for sparse polynomials, in: Proc. EUROSAM 79 (Edward W. Ng, ed.), Lecture Notes in Compu. Sci. 72, Springer-Verlag, Berlin, 1979, 216–226.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jim Geelen.

Additional information

This research was partially supported by a grant from the Office of Naval Research [N00014-10-1-0851].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Geelen, J., Kapadia, R. Computing Girth and Cogirth in Perturbed Graphic Matroids. Combinatorica 38, 167–191 (2018). https://doi.org/10.1007/s00493-016-3445-3

Download citation

Mathematics Subject Classification (2000)

  • 05B35
  • 94B05
  • 90C27