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The Idea is this:

The Decision problems Max Cut for graphs is APX-Hard and therefore does not have a PTAS.

However Dense versions of Max Cut do have a PTAS!

This suggest that Counting Perfect Matchings for Dense Bipartite Graphs could have some sort of speed up.

Does such a speedup exists in the complexity literature?

Further to the above, I am aware of the expectional work done on NonNegative Matrices. However I was really thinking about General matrices and expecting exponential algorithms. Of course #P is by standard complexity theory intractable!

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What about

Jerrum, Mark; Sinclair, Alistair; Vigoda, Eric., A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries, J. ACM 51, No. 4, 671–697 (2004). ZBL1204.65044.

?

If you are looking for exact algorithms, the only speedup I'm aware of is

Bax, E.; Franklin, J., A permanent algorithm with $\exp[\Omega(n^{1/3}/2\ln n)]$ expected speedup for 0-1 matrices, Algorithmica 32, No. 1, 157–162 (2002). ZBL0995.68190.

Which also can be used to show you can do $c^n$ time for $c<2$ for sparse 0/1 matrices or the UP-hard mod 3 permanent.

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