Approximating permanents and hafnians

Abstract

We prove that the logarithm of the permanent of an n×n real matrix A and the logarithm of the hafnian of a 2n×2n real symmetric matrix A can be approximated within an additive error 1 ≥ ε > 0 by a polynomial p in the entries of A of degree O(lnn-lnε) provided the entries aij of A satisfy δ ≤ aij ≤ 1 for an arbitrarily small δ > 0, fixed in advance. Moreover, the polynomial p can be computed in nO(lnn-lnε) time. We also improve bounds for approximating ln per A, ln haf A and logarithms of multi-dimensional permanents for complex matrices and tensors A.