Computing the Permanent of (Some) Complex Matrices.

We present a deterministic algorithm, which, for any given $$0< \epsilon < 1$$ and an $$n \times n$$ real or complex matrix $$A=\left( a_{ij}\right) $$ such that $$\left| a_{ij}-1 \right| \le 0.19$$ for all $$i, j$$ computes the permanent of $$A$$ within relative error $$\epsilon $$ in $$n^{O\left( \ln n -\ln \epsilon \right) }$$ time. The method can be extended to computing hafnians and multidimensional permanents. [ABSTRACT FROM AUTHOR]