Matching is as easy as matrix inversion

Abstract

We present a new algorithm for finding a maximum matching in a general graph. The special feature of our algorithm is that its only computationally non-trivial step is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC 2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show other applications of this lemma to parallel computation and randomized reductions.

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

References

  1. [1]

    S. A. Cook, A Taxonomy of Problems with Fast Parallel Algorithms,Information and Control,64 (1985), 2–22.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    L. Csánky, Fast Parallel Matrix Inversion Algorithms.SIAM J. Computing,5 (1976), 618–623.

    MATH  Article  Google Scholar 

  3. [3]

    J. Edmonds, Paths, Trees and Flowers,Canad. J. Math.,17 (1965), 449–467.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    J. Edmonds, Systems of Distinct Representatives and Linear Algebra,J. Res. Nat. Bureau of Standards, 71B,4 (1967), 241–245.

    MathSciNet  Google Scholar 

  5. [5]

    Z. Galil andV. Pan, Improved Processor Bounds for Algebraic and Combinatorial Problems in RNC,Twenty Sixth Annual IEEE Symp. on the Foundations of Computer Science. (1985), 490–495.

  6. [6]

    H. Karloff, A Randomized Parallel Algorithm for the Odd Set Cover Problem,Combinatorica 6 (1986), 387–391.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    R. M. Karp, E. Upfal andA. Wigderson, Finding a Maximum Matching is in Random NC,Seventeenth Annual Symp. on Theory of Computing. (1985), 22–32.

  8. [8]

    R. M. Karp, E. Upfal andA. Wigderson, Are Search and Decision Problems Computationally Equivalent?Seventeenth Annual Symp. on Theory of Computing. (1985).

  9. [9]

    D. Kozen, U. V. Vazirani andV. V. Vazirani, NC Algorithms for Comparability Graphs, Interval graphs, and Testing for Unique Perfect Matching,Fifth Annual Foundations of Software Technology and Theoretical Computer Science Conference (1985), invited paper inTheoretical Computer Science.

  10. [10]

    L. Lovász, On Determinants, Matchings and Random Algorithms,Fundamentals of Computing Theory, edited by L. Budach, Akademia-Verlag, Berlin, (1979).

    Google Scholar 

  11. [11]

    L. Lovász,Combinatorial Problems and Exercises, Akadémiai Kiadó, and North-Holland, Amsterdam, (1979).

    Google Scholar 

  12. [12]

    L. Lovász andM. Plummer,Matcing Theory, Academic Press, Budapest, Hungary, (1986).

    Google Scholar 

  13. [13]

    S. Micali andV. V. Vazirani, AnO(√|V||E|) Algorithm for Finding Maximum Matching in General Graphs,Twenty First Annual IEEE Symp. on the Foundations of Computer Science (1980), 17–27.

  14. [14]

    V. Pan, Fast and Efficient Algorithms for the Exact Inversion of Integer Matrices,Fifth Annual Foundations of Software Technology and Theoretical Computer Science Conference (1985).

  15. [15]

    C. H. Papadimitriou andM. Yannakakis, The Complexity of Restricted Spanning Tree Problems,Journal of the ACM,29 (1982), 285–309.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    M. O. Rabin andV. V. Vazirani, Maximum Matching in General Gaphs Through Randomization, submitted.

  17. [17]

    J. T. Schwartz, Fast Probabilistic Algorithms for Verification of Polynomial Identities.J. of ACM,27 (1980), 701–717.

    MATH  Article  Google Scholar 

  18. [18]

    W. T. Tutte, The Factorization of Linear Graphs,J. London Math. Soc.,22 (1947), 107–111.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    L. G. Valiant andV. V. Vazirani, NP is as Easy as Detecting Unique Solutions,Seventeenth Annual Symp. on Theory of Computing. (1985), to appear inTheoretical Computer Science.

  20. [20]

    U. V. Vazirani andV. V. Vazirani, The Two-Processor Scheduling Problem is in Random NC,Seventeenth Annual Symp. on Theory of Computing (1985), 11–21, submitted.

  21. [21]

    A. Borodin, S. A. Cook andN. Pippinger, Parallel Computation for Well-endowed Rings and Space Bounded ProbabilisticMachines, Information and Control,58 (1983) 113–136.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Work done while visiting MSRI, Berkeley, in Fall 1985.

Supported by NSF Grant BCR 85-03611 and an IBM Faculty Development Award.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mulmuley, K., Vazirani, U.V. & Vazirani, V.V. Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987). https://doi.org/10.1007/BF02579206

Download citation

AMS subject classification (1980)

  • 68 E 10
  • 05 C 25