Triangular (0,1)-matrices with prescribed row and column sums

Matrices with Prescribed Row and Column Sums

This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.

Boolean matrices with prescribed row/column sums and stable homogeneous polynomials: Combinatorial and algorithmic applications

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. This...

Integral Matrices with Given Row and Column Sums

Let P = (p,,) and Q = (qij) be m x n integral matrices, R and S be integral vectors. Let Nf(R, S) denote the class of all m x n integral matrices A with row sum vector R and column sum vector S satisfying P < A < Q. For a wide variety of classes ‘%$I( R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in @(R, 5). The first charac...

On nonnegative matrices with given row and column sums

Scaling symmetric positive definite matrices to prescribed row sums

We give a constructive proof of a theorem of Marshall and Olkin that any real symmetric positive definite matrix can be symmetrically scaled by a positive diagonal matrix to have arbitrary positive row sums. We give a slight extension of the result, showing that given a sign pattern, there is a unique diagonal scaling with that sign pattern, and we give upper and lower bounds on the entries of ...