Douglas R. Shier

On powers and centers of chordal graphs

A graph is chordal if every cycle of length strictly greater than three has a chord. A necessary and sufficient condition is given for all powers of a chordal graph to be chordal. In addition, it is shown that for connected chordal graphs the center (the set of all vertices with minimum eccentricity) always induces a connected subgraph. A relationship between the radius and diameter of chordal graphs is also established.

Some aspects of perfect elimination orderings in chordal graphs

Abstract This paper studies properties of perfect elimination orderings in chordal graphs. Specific connections to convex subsets and quasiconcave functions in a graph are discussed. Several new schemes for generating all perfect elimination orderings are investigated and related to existing schemes.

Extended dominance and a stochastic shortest path problem

In the context of stochastic networks, we study the problem of finding a path P that combines in a reasonable way the mean m(P) and variance v(P) of its length. Specifically we study a separable objective function that combines these two path measures: namely, z(P)=f(m(P))+g(v(P)), where f is an increasing convex function and g is an increasing concave function. A new type of dominance (e-dominance), stronger than the standard form of dominance, is then introduced, and it is shown to satisfy a certain form of Bellmans optimality principle. This means that it is possible to modify existing label-setting and label-correcting methods by using e-dominance, and without sacrificing optimality. Computational experience with these enhanced labeling algorithms has been promising. Test results for a variety of sample problems show that the e-dominance criterion can often significantly reduce the number of nondominated path vectors, compared to the standard dominance criterion. We observe a consequent reduction in both computation time and storage requirements.

Maximal chordal subgraphs

An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations.

Cut scheduling in the apparel industry

A problem encountered in the apparel industry is that of producing, with no excess, a known number of different styles from the same cloth. This situation occurs, for instance, in the case of special order or made-to-order garments. In the cutting process, plies of cloth are spread on a cutting table, and several patterns are placed across the top ply. Cutting out the patterns through all plies creates a set of bundles of garment pieces, and several such lays may be required to satisfy all demands. The cut scheduling problem concerns finding a feasible cutting schedule having the minimum number of lays. We present an exact enumerative approach that identifies all optimal solutions to a practically important variant of this problem. The availability of multiple solutions allows greater flexibility and permits decision makers to apply additional criteria in selecting an appropriate cutting schedule. Computational evidence shows that our approach can efficiently solve standard test problems from the literature as well as some very challenging examples provided by a global garment manufacturer.

Optimal Locations for a Class of Nonlinear, Single-Facility Location Problems on a Network

This paper investigates a class of single-facility location problems on an arbitrary network. Necessary and sufficient conditions are obtained for characterizing locally optimal locations with respect to a certain nonlinear objective function. This approach produces a number of new results for locating a facility on an arbitrary network, and in addition it unifies several known results for the special case of tree networks. It also suggests algorithmic procedures for obtaining such optimal locations.

Minimum spanning trees in networks with varying edge weights

This paper considers the problem of determining minimum spanning trees in networks in which each edge weight can assume a finite number of distinct values. We use the algebraic structure of an underlying Hasse diagram to describe the relationship between different edge-weight realizations of the network, yielding new results on how MSTs change under multiple edge-weight perturbations. We investigate various implementation strategies for updating MSTs in this manner. Computational results are provided for some challenging test networks.

An Improved Algorithm for Approximating the Performance of Stochastic Flow Networks

A problem encountered in the analysis of communication and other distribution systems is evaluating the performance of the system in meeting user demands with available resources. We consider the case in which user demands and available resources are only known stochastically and connecting links can operate at various levels. This situation can be modeled as a two-terminal stochastic network flow problem, in which each edge of the network assumes a finite number of values (corresponding to different capacity levels) with known probabilities. For each state of the network, we are interested in the maximum demand that can be met using the best allocation of resources. The approach used here to estimate the average unmet demand involves generating only “high leverage” states of the system—states having high probability and/or high values of unmet demand. A new algorithm is proposed for generating such states in monotone order, either by probability or by unmet demand. Bounds on the performance of the networ...

On the Distributed Bellman-Ford Algorithm and the Looping Problem

The classic Bellman-Ford algorithm for calculating shortest paths can be easily adapted to a distributed environment in which the computations are performed locally by identical processors at each network node. A distributed shortest-path algorithm is particularly appropriate for use in communication networks to capitalize on local information rather than rely on a central controller. This paper discusses the behavior of a synchronous version of the distributed Bellman-Ford algorithm in a dynamic environment in which communication link costs can undergo change. Several algorithms are described that mitigate or eliminate the occurrence of looping, which is responsible for degrading the performance of distributed shortest-path algorithms. We provide theoretical and computational evidence to show that two proposed algorithms offer improvements upon the original and modified Bellman-Ford algorithms.

Algebraic methods applied to network reliability problems

An algebraic structure underlying network reliability problems is presented for determining the 2-terminal reliability of directed networks. An iterative algorithm is derived from this algebraic perspective to solve the