Computing permanents of complex diagonally dominant matrices and tensors

We prove that for any λ > 1, fixed in advance, the permanent of an n × n complex matrix, where the absolute value of each diagonal entry is at least λ 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 < ϵ < 1 in quasi-polynomial n O (ln n -ln ϵ ) time. We extend this result to multidimensional permanents of tensors and apply it to weighted counting of perfect matchings in hypergraphs.