Prescribed Row Sums Research Articles

On line sum optimization

We show that the column sum optimization problem, of finding a (0,1)-matrix with prescribed row sums which minimizes the sum of evaluations of given functions at its column sums, can be solved in polynomial time, either when all functions are the same or when all row sums are bounded by any constant. We conjecture that the more general line sum optimization problem, of finding a matrix minimizing the sum of given functions evaluated at its row sums and column sums, can also be solved in polynomial time.

Symmetric and Hankel-symmetric transportation polytopes

In this paper, we consider the symmetric and Hankel-symmetric transportation polytope , which is the convex set of all symmetric and Hankel-symmetric non-negative matrices with prescribed row sum vector R and prescribed column sum vector S. We characterize all extreme points of . Moreover, we show that the extreme points of the polytope of symmetric and Hankel-symmetric doubly stochastic matrices, can be obtained from the extreme points of by specializing to the case that

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion–Hoffman theorem is given.

Construction of all tournament matrices with prescribed row sum vector

In this article, we define the binary codes of tournament matrices in the class T(R) and give unique construction algorithms for matrices which have minimum and maximum binary codes. By introducing a generating algorithm with an order we show that all matrices in class T(R) can be sorted uniquely between matrices with minimum and maximum binary codes.

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 the scaling matrix.

The relationship between the class %plane1D;518;2(R,S) of (0,1,2)-matrices and the collection of constellation matrices

Let %plane1D;518;2(R,S) be the class of all (0, 1, 2)-matrices with a prescribed row sum vector R and column sum vector S. A (0, 1,2)-matrix in %plane1D;518;2(R,S) is defined to be parsimonious provided no (0, 1, 2)-matrix with the same row and column sum vectors has fewer positive entries. In a parsimonious (0, 1, 2)-matrix A there are severe restrictions on the (0, 1)-matrix A(1) which records the positions of the 1's in A. Brualdi and Michael obtained some necessary arithmetic conditions for a set of matrices to serve simultaneously as the 1-pattern matrices for parsimonious matrices in a given class. In this paper, we provide a direct construction that proves that these conditions are also sufficient.

On the existence of sequences and matrices with prescribed partial sums of elements

We prove necessary and sufficient conditions for the existence of sequences and matrices with elements in given intervals and with prescribed lower and upper bounds on the element sums corresponding to the sets of an orthogonal pair of partitions. We use these conditions to generalize known results on the existence of nonnegative matrices with a given zero pattern and prescribed row and column sums. We also generalize recently proven results on the existence of (a real or nonnegative) square matrix A with a given zero pattern and with prescribed row sums such that A + A T is prescribed. We also introduce Hadamard adjustments, by means of which we generalize known results on the scaling of matrices with a given pattern to achieve prescribed row and column sums.

On the existence of sequences and matrices with prescribed partial sums of elements

We prove necessary and sufficient conditions for the existence of sequences and matrices with elements in given intervals and with prescribed lower and upper bounds on the element sums corresponding to the sets of an orthogonal pair of partitions. We use these conditions to generalize known results on the existence of nonnegative matrices with a given zero pattern and prescribed row and column sums. We also generalize recently proven results on the existence of (a real or nonnegative) square matrix A with a given zero pattern and with prescribed row sums such that A + A T is prescribed. We also introduce Hadamard adjustments, by means of which we generalize known results on the scaling of matrices with a given pattern to achieve prescribed row and column sums.

Joint realization of (0,1) matrices revisited

Let R and R′ be nonnegative integral vectors with m components, and let S and S′ be nonnegative integral vectors with n components. Let U ( R, S) denote the class of all (0,1) matrices of size m by n with prescribed row sum vector R and column sum vector S. The classes U ( R, S) and U ( R′, S') are jointly realizable provided there exist matrices A ϵ U ( R, S) and A′ ϵ U ( R′, S′) such that A ⩽ A′. The pair ( R, S) is totally joint provided that for all R′ and S′ the nonemptiness of the three classes U ( R, S), U ( R, S), and U( R′ − R, S′ − S) implies that the classes U ( R, S) and U ( R′, S′) are jointly realizable. Chen and Shastri recently obtained sufficient conditions for the pair (R, S) to be totally joint. However, their conditions involve two forbidden configurations in the matrices in the class U ( R, S) and thus may be difficult to verify. In this paper we present relatively simple conditions on R and S that characterize the two forbidden configurations. As a corollary, we improve the main result of Chen and Shastri.

The class of matrices of zeros, ones, and twos with prescribed row and column sums

We study the class U 2(R,S) of all (0, 1, 2)-matrices with a prescribed row sum vector R and column sum vector S. A (0, 1, 2)-matrix in U 2(R,S) is defined to be parsimonious provided no (0, 1, 2)-matrix with the same row and column sum vectors has fewer positive entries. In a parsimonious (0, 1, 2)-matrix A there are severe restrictions on the (0, 1)-matrix A 〈1〉 which records the positions of the 1's in A. We attempt to understand the relationships between the set of these matrices A 〈1〉 and the pair ( R, S).

Additive decomposition of nonnegative matrices with applications to permanents and scalingt

Let U1 and U2 be compact subsets of m × n nonnegative matrices with prescribed row sums and column sums. Given A in U2 , we study the quantity and the matrices B in U1 that satisfy A−μ(U1;A)B is nonnegative.The quantity is determined. Using the results obtained, we give a lower bound for the permanent of nonnegative matrices. Moreover, we study the scaling parameters of nonnegative matrices. An upper bound and an extremal characterization for their product are given.

The number of symmetric nonnegative integral matrices with prescribed row sums

The number of symmetric nonnegative integral matrices with prescribed row sums

On the extreme points of the polytope of symmetric matrices with given row sums

The extreme points of the convex polytope of nonnegative symmetric matrices of order n with prescribed row sums are fully characterized by their respective graphs. For infinite matrices such a characterization is shown to be impossible. However, after imposing some additional conditions on the positive entries of the matrices, a considerable subfamily of infinite matrices is characterized by its graphs.

On Generalized Tournament Matrices

played (i.e., there are no ties). The outcome of the tournament may be represented by an n x n tournament matrix T where lij = 1 if player i defeats player j (symbolically, i -+ j) and zero otherwise. An n x n matrix P with nonnegative entries is a generalized tournament matrix if P + ptr = J I, where J denotes the matrix of l's and I the identity matrix. We interpret the entry Pij as the a priori probability that player i defeats player j when they meet (we assume Pii = 0 since no one plays against himself). Suppose the association organizing the tournament wishes to devise an equitable system of handicaps so as to neutralize the advantage strong players have over weak players. In ? 3 we discuss two general methods for doing this making use of various properties of generalized tournament matrices developed in ? 2. In the first method, each player bets a certain amount of his own money on each game he plays; in the second method the association awards a certain amount to the winner of each game. The bets and awards for each game are to depend on the two players involved. The association is to determine the amounts of the bets or awards in advance so that every player has the same expected net gain. There will be many fair systems in general and we consider the existence of systems satisfying various additional conditions. There are many ways in which a generalized tournament matrix P might arise; Pij could represent the frequency with which team i has beaten team j in some atheletic competition or Pi, might represent the frequency with which certain consumers prefer item i to itemj in a paired comparison test. One important problem is to determine some reasonable ranking of the n objects and, if possible, to obtain some quantitative measure of their relative merit. Several ranking methods have been proposed (see David [5]). The most obvious method is simply to rank the participants according to the number of games they have won. One well-known method is due to Zermelo, Bradley, Terry and Ford and another to Wei and Kendall. We obtain a new rationale for their methods as special cases of our betting and awards systems. Furthermore, in ? 3.3 we show that a parameter appearing in the Wei-Kendall method provides a natural measure of how evenly the participants are matched. Landau [12] has shown when there exists an ordinary tournament matrix with prescribed row sums; in ? 3.4 we obtain an analogous result for generalized tournament matrices. Some of the results in this paper were announced at the IFIP '68 Congress [161; some of these results were also obtained independently by Daniels [41.

A lower bound for permanents of (0,1)-matrices

Clearly the permanent function is invariant under permutations of rows and columns and under matrix transposition. An up-to-date account of the theory of permanents and an extensive bibliography on the subject is to be found in [2]. The permanent function plays an important part in combinatorics. In fact, the permanent of a (0, 1)-matrix (i.e., a matrix all of whose entries are 0 and 1) is the number of systems of distinct representatives for the corresponding configuration [7, p. 54]. It is therefore of considerable interest to determine the bounds for permanents of (0, 1)-matrices with prescribed row sums (and/or column sums). Let ri and cj denote the ith row sum and the jth column sum of A respectively, i.e.,