Let A be an $n \times n$ matrix with 0-1 valued entries, and let ${\operatorname{per}}(A)$ be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of ${\operatorname{per}}(A)$ and has running time ${\operatorname{poly}}(n)2^{{n / 2}} $, where ${\operatorname{poly}}(n)$ denotes a function that grows polynomially with n.
Let A be an n×n matrix with 0-1 valued entries, and let per(A) be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of per(A) and has running time poly(n)2n/2, where poly(n) denotes a function that grows polynomially with n.
The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws a special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least 1/polyloglog(n). In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices whose module of mean is at least 1/ polylog(n). We note that if the algorithm can be further improved to work with a mean value that is a sufficiently small 1/poly(n), it will disprove a central conjecture for quantum supremacy. Our algorithm is also much simpler and has a better and flexible trade-off for running time. The running time can be quasi-polynomial in both n and 1/∊, or PTAS (polynomial in n but exponential in 1/∊), where ∊ is the approximation parameter.
The matrix permanent belongs to the complexity class #P-Complete. It is generally believed to be computationally infeasible for large problem sizes, and significant research has been done on approximation algorithms for the matrix permanent. We present an implementation and detailed runtime analysis of one such Markov Chain Monte Carlo (MCMC) based Fully Polynomial Randomized Approximation Scheme (FPRAS) for the matrix permanent, which has previously only been described theoretically and with big-Oh runtime analysis. We demonstrate by analysis and experiment that the constant factors hidden by previous big-Oh analyses result in computational infeasibility.
We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero. This question is motivated by the goal of understanding the classical complexity of linear optics and \emph{boson sampling} (Aaronson and Arkhipov '11; Eldar and Mehraban '17). Barvinok's interpolation method enables efficient approximation of the permanent, provided one can establish a sufficiently large zero-free region for the polynomial per(zJ+W), where J is the all-ones matrix and W is a random matrix with independent mean-zero entries.
We show that when the entries of W are standard complex Gaussians, all zeros of the random polynomial per(zJ+W) lie within a disk of radius O~(n−1/3), which yields an approximation algorithm when the bias of the entries is Ω~(n−1/3). Previously, there were no efficient algorithms at biases smaller than 1/polylog(n), and it was unknown whether there typically exist zeros z with ∣z∣≥1. As a complementary result, we show that the bulk of the zeros, namely (1−ε)n of them, have magnitude Θ(n−1/2). This prevents our interpolation method from contradicting the conjectured average-case hardness of approximating the permanent. We also establish analogous zero-free regions for the hardcore model on general graphs with complex vertex fugacities. In addition, we prove universality results establishing zero-free regions for random matrices W with i.i.d. subexponential entries.
We present a randomized algorithm for estimating the permanent of an M×M real matrix A up to an additive error. We do this by viewing the permanent perm(A) of A as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix C. The algorithm outputs the empirical mean SN of this product after sampling N times. Our algorithm runs in total time O(M3+M2N+MN) with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate perm(A) to an additive error of ε(32M∏i=12MCii) in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular C using a semidefinite program and a relation to the Max-Cut problem and cut-norms.
(MATH) We study approximation algorithms for the permanent of an n x n (0,1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1-entry of A independently by ± e, where e is a random basis element of a suitable algebra; then output |det(B)|2. This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions respectively.(MATH) The above result implies that, in principle, this approach gives a fully-polynomial randomized approximation scheme for the permanent, provided |det(B)|2 can be efficiently computed in the Clifford algebras. Since these algebras are non-commutative it is not clear how to do this. However, our second main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al.
The permanent of a matrix has numerous applications but is notoriously hard to compute. While nonnegative matrices admit polynomial approximation schemes based on rapidly mixing Markov chains, the known practical estimators of the permanent rely on importance or rejection sampling. We advance the rejection sampling approach, which provides probabilistic accuracy guarantees, unlike importance sampling. Specifically, we give a novel class of nesting upper bounds and a simple preprocessing method that, in comparison to previous works, enable faster sampling with better acceptance rate; we demonstrate order-of-magnitude improvements with both theoretical and empirical analyses. In addition, we display instances on which our approximation scheme is competitive against state-of-the-art importance sampling based estimators.