Counting magic squares in quasi-polynomial time

We present a randomized algorithm, which, given positive integers n and t and a real number 0< epsilon <1, computes the number Sigma(n, t) of n x n non-negative integer matrices (magic squares) with the row and column sums equal to t within relative error epsilon. The computational complexity of the algorithm is polynomial in 1/epsilon and quasi-polynomial in N=nt, that is, of the order N^{log N}. A simplified version of the algorithm works in time polynomial in 1/epsilon and N and estimates Sigma(n,t) within a factor of N^{log N}. This simplified version has been implemented. We present results of the implementation, state some conjectures, and discuss possible generalizations.Comment: 30 page

Polynomials with Lorentzian Signature, and Computing Permanents via Hyperbolic Programming

We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic polynomials and conic stable polynomials belong to this class, and the set of polynomials with Lorentzian signature is closed. Finally, we develop a method for computing permanents of nonsingular matrices which belong to a class that includes nonsingular $k$-locally singular matrices via hyperbolic programming

An approximation algorithm for counting contingency tables

We present a randomized approximation algorithm for counting contingency tables , m × n non-negative integer matrices with given row sums R = ( r 1 ,…, r m ) and column sums C = ( c 1 ,…, c n ). We define smooth margins ( R , C ) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N O (ln N ) complexity, where N = r 1 + … + r m = c 1 + … + c n . Various classes of margins are smooth, e.g., when m = O ( n ), n = O ( m ) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + ${sqrt{5}}$ )/2 ≈ 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for log-concave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77454/1/20301_ftp.pd

Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U_3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the CKM matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the MNS matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.Comment: 33 pages, 6 figures version 2 - misprints corrected, explicit formulae for phases provide