Hafnians, perfect matchings and Gaussian matrices

TL;DR

This paper studies the Barvinok estimator for hafnians, providing conditions for subexponential accuracy and demonstrating its effectiveness as a polynomial-time approximation for counting perfect matchings in graphs.

Contribution

The paper introduces a near-optimal condition under which the Barvinok estimator achieves subexponential errors for hafnians, enabling efficient approximation of perfect matchings.

Findings

Barvinok estimator achieves subexponential errors under specific conditions.

The condition for accuracy is nearly optimal.

Provides a polynomial-time algorithm for approximate counting of perfect matchings.

Abstract

We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok's estimator gives a polynomial-time algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.