Advertisement

Foundations of Computational Mathematics

, Volume 16, Issue 2, pp 329–342 | Cite as

Computing the Permanent of (Some) Complex Matrices

  • Alexander BarvinokEmail author
Article

Abstract

We present a deterministic algorithm, which, for any given \(0< \epsilon < 1\) and an \(n \times n\) real or complex matrix \(A=\left( a_{ij}\right) \) such that \(\left| a_{ij}-1 \right| \le 0.19\) for all \(i, j\) computes the permanent of \(A\) within relative error \(\epsilon \) in \(n^{O\left( \ln n -\ln \epsilon \right) }\) time. The method can be extended to computing hafnians and multidimensional permanents.

Keywords

Permanent Hafnian Algorithm 

Mathematics Subject Classification

15A15 68C25 68W25 

Notes

Acknowledgments

The author is grateful to the anonymous referees for their careful reading of the paper, useful suggestions and interesting questions.

References

  1. 1.
    S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory of Computing 9 (2013), 143–252.Google Scholar
  2. 2.
    G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A.Marchetti-Spaccamela, and M. Protasi, Complexity and Approximation.Combinatorial Optimization Problems and their Approximability Properties, Springer-Verlag, Berlin 1999.Google Scholar
  3. 3.
    A. Barvinok, Two algorithmic results for the traveling salesman problem, Mathematics of Operations Research 21 (1996), no. 1, 65–84.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Barvinok, Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor, Random Structures & Algorithms 14 (1999), no. 1, 29–61.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Barvinok, Computing the partition function for cliques in a graph, preprint arXiv:1405.1974 (2014).
  6. 6.
    A. Barvinok and A. Samorodnitsky, Computing the partition function for perfect matchings in a hypergraph, Combinatorics, Probability and Computing 20 (2011), no. 6, 815–835.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Barvinok and P. Soberón, Computing the partition function for graph homomorphisms, preprint arXiv:1406.1771 (2014).
  8. 8.
    J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels and R.R. Thomas, The Euclidean distance degree of an algebraic variety, preprint arXiv:1309.0049 (2013).
  9. 9.
    S. Friedland and L. Gurvits, Generalized Friedland-Tverberg inequality: applications and extensions, preprint arXiv:0603410 (2006).
  10. 10.
    M. Fürer, Approximating permanents of complex matrices, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, New York 2000, pp. 667–669.Google Scholar
  11. 11.
    D. Gamarnik and D. Katz, A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix, Journal of Computer and System Sciences 76 (2010), no. 8, 879–883.CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    L. Gurvits, On the complexity of mixed discriminants and related problems, Mathematical Foundations of Computer Science 2005, Lecture Notes in Computer Science, vol. 3618, Springer, Berlin, 2005, pp. 447–458.Google Scholar
  13. 13.
    L. Gurvits and A. Samorodnitsky, Bounds on the permanent and some applications, preprint arXiv:1408.0976 (2014).
  14. 14.
    M. Jerrum, A. Sinclair and E. Vigoda, A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries, Journal of the ACM 51 (2004), no. 4, 671–697.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    N. Linial, A. Samorodnitsky, and A. Wigderson, A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, Combinatorica 20 (2000), no. 4, 545–568.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    H. Minc, Permanents. Encyclopedia of Mathematics and its Applications, Vol. 6, Addison-Wesley Publishing Co., Reading, Mass., 1978.Google Scholar
  17. 17.
    A.D. Scott and A.D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma, Journal of Statistical Physics 118 (2005), no. 5–6, 1151–1261.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    L.G. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8 (1979), no. 2, 189–201.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations