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 < ǫ < 1 in quasi-polynomial nO(lnn−ln ǫ) time. We extend this result to multidimensional permanents of tensors and discuss its application to weighted counting of perfect matchings in hypergraphs.