Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time cn approximation algorithm for the permanent of positive semidefinite matrices where c =
^γ+1
≃ 4:84. We write a natural convex relaxation and show that its optimum solution gives a cn approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices. We also show that our result implies an approximate version of the permanent-ontop conjecture, which was recently refuted in its original form; we show that the permanent is within a cn factor of the top eigenvalue of the Schur power matrix.