Beth Novick

From Hall's Matching Theorem to Optimal Routing on Hypercubes

We introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Halls “marriage” condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(s?n) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabins information disposal algorithm may be improved.

An axiomatization of the median procedure on the n-cube

textabstractThe general problem in location theory deals with functions that find sites on a graph (discrete case) or network (continuous case) in such a way as to minimize some cost (or maximize some benefit) to a given set of clients represented by vertices on the graph or points on the network. The axiomatic approach seeks to uniquely distinguish, by using a list of intuitively pleasing axioms, certain specific location functions among all the arbitrary functions that address this problem. For the median function, which minimizes the sum of the distances to the client locations, three simple and natural axioms, anonymity, betweenness, and consistency suffice on tree networks (continuous case) as shown by Vohra, and on cube-free median graphs (discrete case) as shown by McMorris et.al.. In the latter paper, in the case of arbitrary median graphs, a fourth axiom was added to characterize the median function. In this note we show that, at least for the hypercubes, a special instance of arbitrary median graphs, the above three natural axioms still suffice.

On Combinatorial Properties of Binary Spaces

A binary clutter is the family of inclusionwise minimal supports of vectors of affine spaces over GF(2). Binary clutters generalize various objects studied in Combinatorial Optimization, such as paths, Chinese Postman Tours, multiflows and one-sided circuits on surfaces. The present work establishes connections among three matroids associated with binary clutters, and between any of them and the binary clutter. These connections are then used to compare well-known classes of binary clutters; to provide polynomial algorithms which either confirm the membership in subclasses, or provide a forbidden clutter-minor; to reformulate and generalize a celebrated conjecture of Seymour on ideal binary clutters in terms of multiflows in matroids, and to exhibit new cases of its validity.

A Heuristic Method to Solve the Size Assortment Problem

This paper considers the size assortment problem, in which a large number of size distributions (e.g., for retail stores) need to be aggregated into a relatively small number of groups in an optimal fashion. All stores within a group are then allocated merchandise according to their common size distribution. A neighborhood search heuristic is developed to produce near-optimal solutions. We investigate the use of both random and “intelligent” starting solutions to initiate the heuristic. The intelligent starting solutions are based on efficiently solving one-dimensional versions of the original problem and then combining these into consensus solutions. Computational results are reported for some small specially structured test problems, as well as some large test problems obtained from an industrial client.

Reconfiguration graphs of shortest paths

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An axiomatic approach to location functions on finite metric spaces

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Turán theorems and convexity invariants for directed graphs

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On Ideal Clutters, Metrics and Multiflows

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A tight axiomatization of the median procedure on median graphs

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On irreducible no-hole L(2, 1)-coloring of trees

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