Computing the Permanent of (Some) Complex Matrices

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