Integral matrices with given row and column sums

AbstractLet P = (pij) and Q = (qij) be m × n integral matrices, R and S be integral vectors. Let UPQ(R, S) denote the class of all m × n integral matrices A with row sum vector R and column sum vector S satisfying P ⩽ A ⩽ Q. For a wide variety of classes UPQ(R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in UPQ(R, S). The first characterization unifies the results of Gale-Ryser, Fulkerson, and Anstee. Many other properties of (0, 1)-matrices with prescribed row and column sum vectors generalize to integral classes satisfying the main condition. We also study the decomposibility of integral classes satisfying the main condition. As a consequence of our decomposibility theorem, it follows a theorem of Beineke and Harary on the existence of a strongly connected digraph with given indegree and outdegree sequences. Finally, we introduce the incidence graph for a matrix in an integral class UPQ(R, S) and study the invariance of an element in a matrix in terms of its incidence graph. Analogous to Minty's Lemma for arc colorings of a digraph, we give a very simple labeling algorithm to determine if an element in a matrix is invariant. By observing the relationship between invariant positions of a matrix and the strong connectedness of its incidence graph, we present a very short graph theoretic proof of a theorem of Brualdi and Ross on invariant sets of (0, 1)-matrices. Our proof also implies an analogous theorem for a class of tournament matrices with given row sum vector, as conjectured by the analogy between bipartite tournaments and ordinary tournaments

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

Enumerating contingency tables via random permanents

Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total weight T=T(R, C; W) of non-negative integer matrices (contingency tables) D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C) T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!. In many cases, ln T' provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N x N random matrix with exponentially distributed entries and approximate the expectation by the integral T' of an efficiently computable log-concave function on R^{mn}. Applications to counting integer flows in graphs are also discussed.Comment: 19 pages, bounds are sharpened, references are adde

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

Pseudomoments of the Riemann zeta-function and pseudomagic squares

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix

Monomial integrals on the classical groups

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows to obtain analytical expressions very efficiently. Those expressions contain the matrix dimension as a free parameter.Comment: 16 page

Higher Spin Alternating Sign Matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.Comment: 41 pages; v2: minor change