Approximating the Permanent of a Random Matrix with Vanishing Mean

Abstract

The permanent is #P -hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P -hard to compute on average if we only care about approximation instead of exact computation?In this work we take a first step towards resolving this question: We present a quasipolynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ = O(ln ln n) −1/8 to within inverse polynomial multiplicative error. Alternatively, one can achieve permanent approximation for matrices with mean µ = 1/polylog(n) in time 2 O(n ε ) , for any ε > 0.The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1, 2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note:1. Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean µ.2. Biased Bernoulli: each entry is −1 + µ with probability 1/2, and 1 with probability 1/2.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

“…1 These algorithms have made use of one of two main techniques: decay of correlations, which exploits decreasing in uence of the spins (colors) on distant vertices on the spin at a given vertex; and polynomial interpolation, which uses the absence of zeros of the partition function in a suitable region of the complex plane. Early examples of the decay of correlations approach include [1,2,40], while for early examples of the polynomial interpolation method, we refer to the monograph of Barvinok [3] (see also, e.g., [4,13,25,27,30,34] for more recent examples). Unfortunately, however, in the case of colorings on general bounded degree graphs, these techniques have so far lagged well behind the MCMC algorithms mentioned above.…”

Computing permanents of complex diagonally dominant matrices and tensors

“…In this paper, we continue the line of research started in [Ba16a] and continued, in particular, in [Ba17], [PR17a], [Ba16b], [PR17b], [L+17], [BR17] and [EM17], on constructing efficient algorithms for computing (approximating) combinatorially defined quantities (partition functions) by exploiting the information on their complex zeros. A typical application of the method consists of a) proving that the function in question does not have zeros in some interesting domain in C n and b) constructing a low-degree polynomial approximation for the logarithm of the function in a slightly smaller domain.…”

“…Usually, part a) is where the main work is done: since there is no general method to establish that a multivariate polynomial (typically having many monomials) is non-zero in a domain in C n , some quite clever arguments are being sought and found, cf. [PR17b], [EM17], see also Section 2.5 of [Ba16b] for the very few general results in this respect. Once part a) is accomplished, part b) produces a quasipolynomial approximation algorithm in a quite straightforward way, see Section 2.2 of [Ba16b].…”

Testing for Dense Subsets in a Graph via the Partition Function

“…We got the idea of the proof from [EM17], where a similar question about complex zeros of the permanents of matrices with independent random entries was treated.…”