Kevin R. Hutson

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.

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.

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.

A Neighborhood Search Technique for the Freeze Tag Problem

The Freeze Tag Problem arises naturally in the field of swarm robotics. Given n robots at different locations, the problem is to devise a schedule to activate all robots in the minimum amount of time. Activation of robots, other than the initial robot, only occurs if an active robot physically moves to the location of an inactive robot. Several authors have devised heuristic algorithms to build solutions to the Freeze Tag Problem. Here, we investigate an update procedure based on a hill-climbing, local search algorithm to solve the Freeze-Tag Problem.

Bounding Distributions for the Weight of a Minimum Spanning Tree in Stochastic Networks

This paper considers the problem of determining the distribution of the weight W of a minimum spanning tree for an undirected graph with edge weights that are independently distributed discrete random variables. Using the underlying fundamental cutsets and cycles associated with a spanning tree, we are able to obtain upper and lower bounds on the distribution of W. In turn, these are used to establish bounds on E[W]. Our general method for deriving these bounding distributions subsumes existing approximation methods in the literature. Computational results indicate that the new approximation methods provide excellent bounds for some challenging test networks.

The Expected Length of a Minimal Spanning Tree of a Cylinder Graph

A cylinder graph is the graph Cartesian product of a path and a cycle. In this paper we investigate the length of a minimal spanning tree of a cylinder graph whose edges are assigned random lengths according to independent and uniformly distributed random variables. Our work was inspired by a formula of J. Michael Steele which shows that the expected length of a minimal spanning tree of a connected graph can be calculated through the Tutte polynomial of the graph. First, using transfer matrices, we show how to calculate the Tutte polynomials of cylinder graphs. Second, using Steeles formula, we tabulate the expected lengths of the minimal spanning trees for some cylinder graphs. Third, for a fixed cycle length, we show that the ratio of the expected length of a minimal spanning tree of a cylinder graph to the length of the cylinder graph converges to a constant; this constant is described in terms of the Perron–Frobenius eigenvalue of the accompanying transfer matrix. Finally, we show that the length of a minimal spanning tree of a cylinder graph satisfies a strong law of large numbers.

Improving the quality of the assignment of students to first-year seminars

Many post-secondary academic institutions in the United States have a First-Year Seminar Program. These seminars are designed to support the success of new incoming first-year students by combining writing, research and active discussion among small groups of students. At Dickinson College, students are required to select six seminars they find interesting from a list of approximately 42 seminars. The college then attempts to assign each student to a seminar on their list, while maintaining course capacities. Using standard commercial optimization software, we develop an approach that not only solves this basic assignment problem, but also seeks to balance both the gender and number of international students in the seminars. In addition, we utilize Monte Carlo simulation to study how the number of seminars each student is required to select affects the likelihood that a feasible assignment exists.

Optimal online ring routing

We study how to route online splittable flows in bidirectional ring networks to minimize maximum load. We show that the competitive ratio of any deterministic online algorithm for this problem is at least

A Comparison of Algorithms for Finding an Efficient Theme Park Tour

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Identifying influential edges in a directed network: big events, upsets and non-transitivity

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