# Approximating the Permanent

@article{Jerrum1989ApproximatingTP, title={Approximating the Permanent}, author={Mark Jerrum and Alistair Sinclair}, journal={SIAM J. Comput.}, year={1989}, volume={18}, pages={1149-1178} }

A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class…

## 799 Citations

### A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 2010

### Polynomial-Time Approximation of the Permanent

- Mathematics
- 2011

Despite its apparent similarity to the (easily-computable) determinant, it is believed that there is no polynomial-time algorithm for computing the permanent of an arbitrary matrix. In this survey,…

### An analysis of a Monte Carlo algorithm for estimating the permanent

- MathematicsIPCO
- 1993

It is shown that polynomially many trials suffice to approximate the permanent of any dense 0,1-matrix, i.e., one in which every row- and column-sum is at least (1/2+α)n, for some constant α>0.

### An analysis of a random algorithm for estimating all the matchings

- Computer ScienceArXiv
- 2007

A simple approach (RM) to approximate the permanent, which just yields a critical ratio O($n\omega(n)$) for almost all the 0-1 matrices, provided it's a simple promising practical way to compute this #P-complete problem.

### An Almost Linear Time Approximation Algorithm for the Permanen of a Random (0-1) Matrix

- Computer Science, MathematicsFSTTCS
- 2004

The algorithm with inputs A, ∈ > 0 produces an output X A with (1-∈)per(A) 0, and almost all (0-1) matrices the algorithm runs in time O(n 2 ω), i.e., almost linear in the size of the matrix.

### Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

- Computer ScienceCombinatorics, Probability and Computing
- 1992

A new upper bound on the mixing rate is presented, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph, and improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system.

### Fast approximation of the permanent for very dense problems

- Computer ScienceSODA '08
- 2008

This work presents a very different approach using sequential acceptance/rejection to approximation of the permanent of a matrix with nonnegative entries, and shows that for a class of dense problems this method has an O(n)(log 4) expected running time.

### A Decomposition Based Proof for Fast Mixing of a Markov Chain over Balanced Realizations of a Joint Degree Matrix

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2015

It is proved that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes.

### Approximating permanents of complex matrices

- Computer ScienceSTOC '00
- 2000

This work presents the first approximation algorithm for the permanent of an arbi trary complex matrix, and extends the notion of an (e,6)-approximation algorithm to accommodate for cancellations in additions.

### Approximating the Permanent by Sampling from Adaptive Partitions

- Computer ScienceNeurIPS
- 2019

ADAPART uses an adaptive, iterative partitioning strategy over permutations to convert any upper bounding method for the permanent into one that satisfies a desirable `nesting' property over the partition used, and provides significant speedups over prior work.

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