Advertisement

On the linear classification of even and odd permutation matrices and the complexity of computing the permanent

Article
  • 30 Downloads

Abstract

The problem of linear classification of the parity of permutation matrices is studied. This problem is related to the analysis of complexity of a class of algorithms designed for computing the permanent of a matrix that generalizes the Kasteleyn algorithm. Exponential lower bounds on the magnitude of the coefficients of the functional that classifies the even and odd permutation matrices in the case of the field of real numbers and similar linear lower bounds on the rank of the classifying map for the case of the field of characteristic 2 are obtained.

Keywords

permutation matrix parity permanent linear classification theta function independence number 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Abbe, N. Alon, and A. S. Bandeira, “Linear Boolean classification, coding and the critical problem”, arXiv:1401.6528v2. 2014.Google Scholar
  2. 2.
    H. H. Crapo and G. C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries (MIT, Cambridge, Mass., 1970).MATHGoogle Scholar
  3. 3.
    M. Aigner, Combinatorial Theory (Springer, Berlin, 1997; Mir, Moscow, 1982).CrossRefMATHGoogle Scholar
  4. 4.
    S. Arora and B. Barak, Computational Complexity: A Modern Approach (Cambridge Univ. Press, Cambridge, 2009).CrossRefMATHGoogle Scholar
  5. 5.
    L. G. Valiant, “The complexity of computing the permanent,” Theor. Comput. Sci. 8 (2), 189–201 (1979).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H. J. Ryser, Combinatorial Mathematics, Carus Math. Monographs, No. 14 (Math. Assoc. of America, Washington, 1963).Google Scholar
  7. 7.
    M. Mahajan and V. Vinay, “Determinant: old algorithms, new insights,” SIAM J. Discr. Math. 12, 474–490 (1999).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P. W. Kasteleyn, “Graph theory and crystal physics,” in Graph Theory and Theoretical Physics, Ed. by F. Harary (Academic, London, 1967), pp. 43–110.Google Scholar
  9. 9.
    Yu. G. Smetanin and L. G. Khachiyan, “Application of pseudopolynomial algorithms to some combinatorial constrained optimization problems,” Izv. Akad. Nauk. SSSR, Tekh. Kibern., No. 6, 139–144 (1986).Google Scholar
  10. 10.
    L. Lovasz, “Semidefinite programs and combinatorial optimization,” in Recent Advances in Algorithms and Combinatorics, Ed. by B. A. Reed and C. L. Sales (Springer, Berlin, 2003), pp. 137–194.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgopudnyi, Moscow oblastRussia
  2. 2.Dorodnicyn Computing CenterFederal Research Center “Computer Science and Control,” Russian Academy of SciencesMoscowRussia

Personalised recommendations