On the Hardness of Permanent

  • Jin-Yi Cai
  • A. Pavan
  • D. Sivakumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1563)

Abstract

We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P#p = BPP. Our algorithm works over any sufficiently large finite field (polynomially larger than the inverse of the assumed success ratio), or any interval of integers of similar range. The assumed algorithm can also be a probabilistic polynomial time algorithm. Our result is essentially the best possible based on any black box assumption of permanent solvers, and is a simultaneous improvement of the results of Gemmell and Sudan [GS92], Feige and Lund [FL92] as well as Cai and Hemachandra [CH91], and Toda (see [ABG90]).

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References

  1. [ABG90]
    A. Amir, R. Beigel, and W. Gasarch. Some connections between query classes and non-uniform complexity. In Proceedings of the 5th Structure in Complexity Theory, pages 232–243. IEEE Computer Society, 1990.Google Scholar
  2. [ALRS92]
    S. Ar, R. Lipton, R. Rubinfeld, and M. Sudan. Reconstructing algebraic functions from mixed data. In Proc. 33rd FOCS, pages 503–512, 1992.Google Scholar
  3. [BW]
    E. Berlekamp and L. Welch. Error correction of algebraic codes. US Patent Number 4,633,470.Google Scholar
  4. [CH91]
    J. Cai and L. Hemachandra. A note on enumerative counting. Information Processing Letters, 38(4):215–219, 1991.MATHCrossRefMathSciNetGoogle Scholar
  5. [FL92]
    U. Feige and C. Lund. On the hardness of computing permanent of random matrices. In Proceedings of 24th STOC, pages 643–654, 1992.Google Scholar
  6. [GLRSW91]
    P. Gemmell, R. Lipton, R. Rubinfeld, M. Sudan, and A. Wigderson. Self-testing/correcting for polynomials and for approximate functions. In Proceedings of 23rd STOC, pages 32–42, 1991.Google Scholar
  7. [GS92]
    P. Gemmell and M. Sudan. Highly resilient correctors for polynomials. Information Processing Letters, 43:169–174, 1992.MATHCrossRefMathSciNetGoogle Scholar
  8. [GRS98]
    O. Goldreich and D. Ron and M. Sudan. Chinese remaindering with errors. ECCC Technical Report TR 98-062, October 29, 1998. Available at http://www.eccc.uni-trier.de.
  9. [IW98]
    R. Impagliazzo and A. Wigderson. Randomness vs Time, Derandomization under a uniform assumption. Manuscript, 1998. To appear in FOCS’ 98.Google Scholar
  10. [Kal92]
    E. Kaltofen. Polynomial factorization 1987–1991. LATIN’ 92, I. Simon (Ed.), LNCS, vol. 583, pp294–313, Springer, 1992.CrossRefGoogle Scholar
  11. [LFKN90]
    C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. In Proceedings of 31st FOCS, pages 2–10, 1990.Google Scholar
  12. [Lip91]
    R. Lipton. New directions in testing, In Distributed Computing and Cryptography, volume 2 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 191–202. AMS, 1991Google Scholar
  13. [MR95]
    R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.Google Scholar
  14. [Rys63]
    H. J. Ryser. Combinatorial Mathematics. Carus Mathematical Monograph No 14, Math. Assoc. of America, 1963.Google Scholar
  15. [Sud96]
    M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proceedings of the 37th FOCS, pages 164–172, 1996.Google Scholar
  16. [Tod89]
    S. Toda. On the computational power of PP and ◯P. In Proceedings of the 30th FOCS, pages 514–519, 1989.Google Scholar
  17. [Val79]
    L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 47(1):85–93, 1979.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • A. Pavan
    • 1
  • D. Sivakumar
    • 2
  1. 1.Department of Computer Science and EngineeringState University of New York at BuffaloBuffalo
  2. 2.Department of Computer ScienceUniversity of HoustonHouston

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