On the hardness of approximating the permanent of structured matrices

Authors:
Bruno Codenotti;Igor E. Shparlinski;Arne Winterhof
Affiliations:
Department of Computer Science, University of Iowa, 14 MacLean Hall, Iowa City, IA;Department of Computing, Macquarie University, Sydney, NSW 2109, Australia;Department of Computer Science, University of Chicago, Ryerson Hall, 1100E 58th Street, Chicago, IL
Venue:
Computational Complexity
Year:
2003

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Abstract

We show that for several natural classes of "structured" matrices, including symmetric, circulant, Hankel and Toeplitz matrices, approximating the permanent modulo a prime p is as hard as computing its exact value. Results of this kind are well known for arbitrary matrices. However the techniques used do not seem to apply to "structured" matrices. Our approach is based on recent advances in the hidden number problem introduced by Boneh and Venkatesan in 1996 combined with some bounds of exponential sums motivated by the Waring problem in finite fields.