# Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

## TL;DR

This paper introduces a deterministic polynomial-time algorithm that approximates the permanent of positive semidefinite matrices within a factor of approximately 4.84, using a convex relaxation approach.

## Contribution

It presents the first polynomial-time approximation algorithm with a provable approximation factor for the permanent of positive semidefinite matrices.

## Key findings

- The algorithm achieves a $c^n$ approximation with $c \\approx 4.84$.
- The convex relaxation used is natural and effective.
- The approximation factor is shown to be asymptotically tight.

## Abstract

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{\gamma+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.