On the Hardness of Permanent
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]).
KeywordsSuccess Probability Polynomial Time Algorithm Success Ratio Permanent Function Bernoulli Trial
Unable to display preview. Download preview PDF.
- [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
- [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
- [BW]E. Berlekamp and L. Welch. Error correction of algebraic codes. US Patent Number 4,633,470.Google Scholar
- [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
- [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
- [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.
- [IW98]R. Impagliazzo and A. Wigderson. Randomness vs Time, Derandomization under a uniform assumption. Manuscript, 1998. To appear in FOCS’ 98.Google Scholar
- [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
- [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
- [MR95]R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.Google Scholar
- [Rys63]H. J. Ryser. Combinatorial Mathematics. Carus Mathematical Monograph No 14, Math. Assoc. of America, 1963.Google Scholar
- [Sud96]M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proceedings of the 37th FOCS, pages 164–172, 1996.Google Scholar
- [Tod89]S. Toda. On the computational power of PP and ◯P. In Proceedings of the 30th FOCS, pages 514–519, 1989.Google Scholar