# Approximating permanents and hafnians

@article{Barvinok2016ApproximatingPA, title={Approximating permanents and hafnians}, author={Alexander I. Barvinok}, journal={arXiv: Combinatorics}, year={2016} }

We prove that the logarithm of the permanent of an nxn real matrix A and the logarithm of the hafnian of a 2nx2n real symmetric matrix A can be approximated within an additive error 1 > epsilon > 0 by a polynomial p in the entries of A of degree O(ln n - ln epsilon) provided the entries a_ij of A satisfy delta 0, fixed in advance. Moreover, the polynomial p can be computed in n^{O(ln n - ln epsilon)} time. We also improve bounds for approximating ln per A, ln haf A and logarithms of multi…

## 23 Citations

### Computing permanents of complex diagonally dominant matrices and tensors

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- 2019

We prove that for any $\lambda > 1$, fixed in advance, the permanent of an $n \times n$ complex matrix, where the absolute value of each diagonal entry is at least $\lambda$ times bigger than the sum…

### Approximating the Permanent of a Random Matrix with Vanishing Mean

- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

A quasi-polynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ to within inverse polynomial multiplicative error to counter the common intuition that the difficulty of computing the permanent stems merely from the authors' inability to treat matrices with many opposing signs.

### A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

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The condition number of a matrix implies an upper bound on the depth complexity (and therefore also on the space complexity) for this task: the determinant of Hermitian matrices with condition number $\kappa$ can be approximated to inverse polynomial relative error with classical circuits of depth.

### Angle-Restricted Sets and Zero-Free Regions for the Permanent

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A systematic method of constructing zero-free regions for the permanent in the sense of A, i.e. regions in the complex plane such that the permanent of a square matrix of any size with entries from this region is nonzero is given.

### Spectral Independence via Stability and Applications to Holant-Type Problems

- Mathematics, Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
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This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo algorithms, and obtains optimal mixing time bounds for the single-site update Markov chain known as the Glauber dynamics.

### Zero-Free Regions of Partition Functions with Applications to Algorithms and Graph Limits

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A class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs is identified and a quasi-polynomial time approximation scheme for computing these partition functions is given.

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