  We prove that for any $\lambda > 1$, fixed in advance, the permanent of an $n
\times n$ complex matrix, where the absolute value of each diagonal entry is at
least $\lambda$ times bigger than the sum of the absolute values of all other
entries in the same row, can be approximated within any relative error $0 <
\epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon)}$ time. We extend
this result to multidimensional permanents of tensors and discuss its
application to weighted counting of perfect matchings in hypergraphs.
