Advertisement

On the permanent of certain circulant matrices

  • B. Codenotti
  • G. Resta

Abstract

The computation of the permanent of a matrix seems to be a very hard task, even for sparse (0, 1)-matrices. A number of results show that it is extremely unlikely that there is a polynomial time algorithm for computing the permanent. The best known algorithm is due to Ryser [22] and takes O(n 2 n ) operations, where n is the matrix size.

Keywords

Bipartite Graph Planar Graph Adjacency Matrix Polynomial Time Algorithm Toeplitz Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bernasconi, A., Codenotti, B., Crespi, V., Resta, G. (1999): How fast can one compute the permanent of circulant matrices? Linear Algebra Appl. 292, 15–37MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bose, R.C., Chowla, S. (1962): Theorems in the additive theory of numbers. Comment. Math. Helv. 37, 141–147MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Brualdi, R.A, Shader, B.L. (1995): Matrices of sign-solvable linear systems. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  4. [4]
    Codenotti, B., Crespi, V., Resta, G. (1997): On the permanent of certain (0, 1) Toeplitz matrices. Linear Algebra Appl. 267, 65–100MathSciNetMATHGoogle Scholar
  5. [5]
    Codenotti, B., Resta, G. (2000): Circulant permanents. In preparationGoogle Scholar
  6. [6]
    Dagum, R, Luby, M., Mihail, M., Vazirani, U. (1988): Polytopes, permanents, and graphs with large factors. In: Proceedings of the 29th IEEE Symposium on Foundations of Computer Science. Institute of Electrical and Electronics Engineers. Washington, DC, pp. 412–421Google Scholar
  7. [7]
    Dietrich, C.R., Osborne, M.R. (1996): O (n log2 n) determinant computation of aToeplitz matrix and fast variance estimation. Appl. Math. Lett. 9, 29–31MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Elspas, B., Turner, J. (1970): Graphs with circulant adjacency matrices. J. Combin. Theory 9, 297–307MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Feige, U., Lund, C. (1992): On the hardness of computing the permanent of random matrices. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on the Theory of Computing. Association for Computing Machinery, New York, pp. 643–654Google Scholar
  10. [10]
    Galluccio, A., Loebl, M. (1999): On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin. 6, R6MathSciNetGoogle Scholar
  11. [11]
    Galluccio, A., Loebl, M. (1999): On the theory of Pfaffian orientations. II. T-joins, k-cuts, and duality of enumeration. Electron. J. Combin. 6, R7MathSciNetGoogle Scholar
  12. [12]
    Halberstam, H., Roth, K.F. (1966): Sequences. Clarendon Press, OxfordMATHGoogle Scholar
  13. [13]
    Kac, M., Ward, J.C. (1952): A combinatorial solution of the two-dimensional Ising model. Phys. Rev. 88, 1332–1337MATHCrossRefGoogle Scholar
  14. [14]
    Kasteleyn, P.W. (1961): The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225MATHCrossRefGoogle Scholar
  15. [15]
    Kasteleyn, P.W. (1963): Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293MathSciNetCrossRefGoogle Scholar
  16. [16]
    Kasteleyn, P.W. (1967): Graph theory and crystal physics. In: Harary, F. (ed.). Graph Theory and Theoretical Physics. Academic Press, London, pp. 43–110Google Scholar
  17. [17]
    Lovasz, L., Plummer, M.D. (1986): Matching theory. (Annals of Discrete Mathematics, vol. 29). North-Holland, AmsterdamGoogle Scholar
  18. [18]
    McCuaig, W., Robertson, N., Seymour, P.D., Thomas, R. (1997): Permanents, Pfaffian orientations, and even directed circuits (extended abstract). Proceedings of the Twenty-Ninth ACM Symposium on the Theory of Computing. Association for Computing Machinery, New York, pp. 402–405Google Scholar
  19. [19]
    Minc, H. (1978): Permanents. (Encyclopedia of Mathematics and its Applications, vol. 6). Addison-Wesley, Reading, MAMATHGoogle Scholar
  20. [20]
    Minc, H. (1987): Permanental compounds and permanents of (0, 1) circulants. Linear Algebra Appl. 86, 11–42MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Polya, G. (1913): Aufgabe 424. Arch. Math. Phys. (3), 20, 271Google Scholar
  22. [22]
    Ryser, H.J. (1963): Combinatorial mathematics. (The Carus Mathematical Monographs no. 14). Mathematical Association of America, Buffalo, NYMATHGoogle Scholar
  23. [23]
    Valiant, L.G. (1979): The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189–201MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Valiant, L.G. (1979): Completeness classes in algebra. In: Conference Record of the Eleventh ACM Symposium on Theory of Computing. Association for Computing Machinery, New York, pp. 249–261Google Scholar

Copyright information

© Springer-Verlag Italia 2001

Authors and Affiliations

  • B. Codenotti
  • G. Resta

There are no affiliations available

Personalised recommendations