For PSD matrices with no zero diagonal, per(A) satisfies e^{-γ n}widehat{P}(A) ≤ per(A) ≤ widehat{P}(A) where widehat{P} is obtained by maximizing a concave function, giving an optimal e^{(γ+o(1))n} deterministic approximation.
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Optimal $e^{(\gamma+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices
For PSD matrices with no zero diagonal, per(A) satisfies e^{-γ n}widehat{P}(A) ≤ per(A) ≤ widehat{P}(A) where widehat{P} is obtained by maximizing a concave function, giving an optimal e^{(γ+o(1))n} deterministic approximation.