Advertisement

Pfaffian orientations, 0/1 permanents, and even cycles in directed graphs

  • Vijay V. Vazirani
  • Mihalis Yannakakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)

Abstract

The following issues in computational complexity remain imprecisely understood: the striking difference in the complexities of computing the permanent and determinant of a matrix despite their similar looking formulae, the complexity of checking if a directed graph contains an even length cycle, and the complexity of computing the number of perfect matchings in a graph using Pfaffian orientations. Via polynomial time equivalences, we show interrelationships among these issues.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Be]
    C. Berge, Graphs and Hypergraphs, North Holland Publishing Co., Amsterdam, 1973.Google Scholar
  2. [Br]
    R. A. Brualdi, ‘Counting Permutations with Restricted Positions: Permanents of (0,1)-Matrices,’ to appear in Lin. Algebra and Appl.Google Scholar
  3. [Fr]
    S. Friedland, ‘Every 7-Regular Digraph Contains an Even Cycle,’ to appear.Google Scholar
  4. [KUW]
    R. M. Karp, E. Upfal, and A. Wigderson, ‘Constructing a Maximum Matching is in Random NC,’ Combinatorica, 6(1), (1986), 35–48.Google Scholar
  5. [Ka]
    P. W. Kasteleyn, ‘Graph Theory and Crystal Physics,’ Graph Theory and Theoretical Physics, Ed.: F. Harary, Academic Press, NY (1967), 43–110.Google Scholar
  6. [KLM]
    V. Klee, R. Ladner, and R. Manber, ‘Sign solvability revisited.’ Lin. Algebra and Appl. 59 (1984), 131–158.Google Scholar
  7. [Ko]
    K.M. Koh, ‘Even Circuits in Directed Graphs and Lovasz's Conjecture,’ Bull. Malaysian Math. Soc. 7 (1976), 47–52.Google Scholar
  8. [Li]
    C. H. C. Little, An extension of Kasteleyn's method of enumerating the 1-factors of planar graphs, Combinatorial Mathematics, Proc. Second Australian Conference, Ed.: D. Holton, Lecture Notes in Math. 403, Springer-Verlag Berlin, (1974), 63–72.Google Scholar
  9. [L1]
    L. Lovász, Problem 2, in Recent Advances in Graph Theory (M. Fiedler, Ed.), Proc. Symp Prague 1974, Academia Praha, Prague, 1975.Google Scholar
  10. [L2]
    L. Lovász, ‘Matching Structure and the Matching Lattice,’ J. Comb. Th., Series B, 43 (1987), 187–222.Google Scholar
  11. [LP]
    L. Lovász and M. Plummer, Matching Theory, Academic Press, Budapest, Hungary.Google Scholar
  12. [MS]
    R. Manber and J. Shao, ‘On Digraphs with the Odd Cycle Property', Journal of Graph Theory, vol. 10 (1986), 155–165.Google Scholar
  13. [MM]
    M. Marcus and H. Minc, ‘On the relation between the determinant and the permanent,’ Illinois J. Math. 5: 376–381 (1961).Google Scholar
  14. [MVV]
    K. Mulmuley, U.V. Vazirani, and V. V. Vazirani, ‘Matching is as easy as matrix multiplication,’ Combinatorica 7:1 (1987), 105–113.Google Scholar
  15. [Po]
    G. Polya, Aufgabe 424, Arch. Math. Phys. (3) 20: 271 (1913).Google Scholar
  16. [ST-]
    P. Seymour and C. T. Thomassen, ‘Characterization of even directed graphs,’ J. Comb. Th., Series B, 42 (1987), 36–45.Google Scholar
  17. [T1]
    C. Thomassen, ‘Even cycles in directed graphs.’ European J. Comb. 6 (1985), 85–89.Google Scholar
  18. [T2]
    C. Thomassen, ‘Sign-nonsingular matrices and even cycles in directed graphs.’ Lin.Algebra and Appl. 75 (1986), 27–41.Google Scholar
  19. [Va]
    L. G. Valiant, ‘The Complexity of Computing the Permanent,’ Theoretical Computer Science 8 (1979), 189–201.Google Scholar
  20. [VV]
    L. G. Valiant and V. V. Vazirani, ‘NP is as Easy as Detecting Unique Solutions,’ Theoretical Computer Science, 47 (1986), 85–93.Google Scholar
  21. [V]
    V. V. Vazirani, ‘NC Algorithms for Computing the Number of Perfect Matchings in K 3,3-Free Graphs and Related Problems,’ submitted for publication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Vijay V. Vazirani
    • 1
  • Mihalis Yannakakis
    • 2
  1. 1.Computer Science DepartmentCornell UniversityUSA
  2. 2.AT&T Bell LabsMurray Hill

Personalised recommendations