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.
Similar content being viewed by others
References
E. Abbe, N. Alon, and A. S. Bandeira, “Linear Boolean classification, coding and the critical problem”, arXiv:1401.6528v2. 2014.
H. H. Crapo and G. C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries (MIT, Cambridge, Mass., 1970).
M. Aigner, Combinatorial Theory (Springer, Berlin, 1997; Mir, Moscow, 1982).
S. Arora and B. Barak, Computational Complexity: A Modern Approach (Cambridge Univ. Press, Cambridge, 2009).
L. G. Valiant, “The complexity of computing the permanent,” Theor. Comput. Sci. 8 (2), 189–201 (1979).
H. J. Ryser, Combinatorial Mathematics, Carus Math. Monographs, No. 14 (Math. Assoc. of America, Washington, 1963).
M. Mahajan and V. Vinay, “Determinant: old algorithms, new insights,” SIAM J. Discr. Math. 12, 474–490 (1999).
P. W. Kasteleyn, “Graph theory and crystal physics,” in Graph Theory and Theoretical Physics, Ed. by F. Harary (Academic, London, 1967), pp. 43–110.
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).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Babenko, M.N. Vyalyi, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 2, pp. 362–372.
Rights and permissions
About this article
Cite this article
Babenko, A.V., Vyalyi, M.N. On the linear classification of even and odd permutation matrices and the complexity of computing the permanent. Comput. Math. and Math. Phys. 57, 362–371 (2017). https://doi.org/10.1134/S0965542517020038
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517020038