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]).
Supported in part by NSF grant CCR-9634665, and by a Guggenheim Fellowship.
Supported in part by an NSF CAREER award CCR-9734164.
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Cai, JY., Pavan, A., Sivakumar, D. (1999). On the Hardness of Permanent. In: Meinel, C., Tison, S. (eds) STACS 99. STACS 1999. Lecture Notes in Computer Science, vol 1563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49116-3_8
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DOI: https://doi.org/10.1007/3-540-49116-3_8
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