Uriel G. Rothblum

Nonnegative Ranks, Decompositions, and Factorizations of Nonnegative Matrices

The nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively. Such decompositions are useful in diverse scientific disciplines. We obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm.

Stable matchings and linear inequalities

The theory of linear inequalities and linear programming was recently applied to study the stable marriage problem which until then has been studied by mostly combinatorial methods. Here we extend the approach to the general stable matching problem in which the structure of matchable pairs need not be bipartite. New issues arise in the analysis and we combine linear algebra and graph theory to explore them.

Paths to marriage stability

We obtain a family of algorithms that determine stable matchings for the stable marriage problem by starting with an arbitrary matching and iteratively satisfying blocking pairs, that is, matching couples who both prefer to be together over the outcome of the current matching. The existence of such an algorithm is related to a question raised by Knuth (1976) and was recently resolved positively by Roth and Vande Vate (1992). The basic version of our method depends on a fixed ordering of all mutually acceptable man-woman pairs which is consistent with the preferences of either all men or of all women. Given such an ordering, we show that starting with an arbitrary matching and iteratively satisfying the highest blocking pair at each iteration will eventually yield a stable matching. We show that the single-proposal variant of the Gale-Shapley algorithm as well as the Roth-Vande Vate algorithm are instances of our approach. We also demonstrate that an arbitrary decentralized system does not guarantee convergence to a stable matching.

Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices

Let A and B be two n×n complex matrices, and let λ be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form λ+σ∞k=1λkek and σ∞k=0ʋkek, respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A+eB for e that has sufficiently small absolute value. Our results apply to the case where λ is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and λ is the spectral radius of A. In particular, if A+eB is nonnegative for sufficiently small nonnegative e and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A+eB and for corresponding eigenvectors. The coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix. Our approach is constructive, and we present a recursive algorithm that will compute the coefficients of the above series.

Courtship and linear programming

This paper demonstrates that the celebrated Gale-Shapley algorithm for obtaining stable matchings in stable marriage problems is essentially an application of the dual simplex method.

THE OPTIMALITY OF THE “CUT ACROSS THE BOARD” RULE APPLIED TO AN INVENTORY MODEL

A common procedure in budget allocation is to let the different entities of an organization determine their optimal budgets. Once the individual requests are received, they are then cut by a common factor, as necessary, so that a global constraint is satisfied. We refer to this procedure as the “cut across the board” rule. In general, this method will not result in a globally optimal solution. In this paper we identify conditions that assure the global optimality of die “cut (or expand) across the board” rule. We specifically focus on a constrained multi-item inventory model and generalize results of Rosenblatt [10] and Plossl and Wight [8]. In addition, we briefly discuss applicability of the results to other areas.

Characterizations of max-balanced flows

Abstract Let G=(V,A) be a graph with vertex set V and arc set A. A flow for G is an arbitrary real-valued function defined on the arcs A. A flow f is called max-balanced if for every cut W,∅≠W⊂V, the maximum flow over arcs leaving W equals the maximum flow over arcs entering W. We describe ten characterizations of max-balanced flows using properties of graph contractions, maximum cycle means, flow maxima, level sets of flows, cycle covers, and minimality with respect to order structure in the set of flows derived from a given flow by reweighting. We also give a linear programming based proof for an existence result of Schneider and Schneider.

The Pareto set of the partition bargaining problem

Abstract We consider games where items are partitioned between two individuals having additive utilities for bundles. Modeling the problem as a bargaining game, we show that all solutions that satisfy the Pareto-optimality axiom assign to one player items with high ratios of the two utilities while the other gets items with low ratios. At most one item is assigned by a lottery. Further, we develop an efficient method for computing the Nash solution (and other popular solutions) for the partitioning problem. We illustrate our results on a partitioning problem introduced in Nashs original paper which he solved for a degenerate instance.

INEQUALITIES OF RAYLEIGH QUOTIENTS AND BOUNDS ON THE SPECTRAL RADIUS OF NONNEGATIVE SYMMETRIC MATRICES

Abstract Given a square, nonnegative, symmetric matrix A, the Rayleigh quotient of a nonnegative vector u under A is given by Q A (u) = u T Au u T u . We show that QA(√u°Au) is not less than QA(u), where √ denotes coordinatewise square roots and ° is the Hadamard product, but that QA(Au) may be smaller than QA(u). Further, we examine issues of convergence.

Some comments on the optimal assembly problem

It is shown that two recent results of Baxter and Harche [1] on monotone and balanced optimal assemblies hold only under conditions that are more restrictive than those originally proposed by the authors. We describe such additional conditions, illustrate why they are needed, and establish their sufficiency. We also consider a recent result by Malon [11] and demonstrate that, while the result itself is correct, its two proofs were incomplete. A complete proof of an extension of the result is then suggested.