Saarbrücken, Univ., Diss., 2003. / Computerdatei im Fernzugriff.

Mannheim, University, Diss., 2004. / Erscheinungsjahr an der Haupttitelstelle: 2003.

Thesis (doctoral)--Technische Hochschule Fridericiana (Karlsruhe, Germany), 2001.

In this paper, we describe the facial structure of the steiner problem in a directed graph by formulating it as a set covering problem. We first characterize trivial facets and derive a necessary condition for nontrivial facets. We also introduce a class of valid inequalities with 0-1 coefficients and show when such inequalities define facets.

University, Diss., 2004--Bonn.

University, Diss., 2003--Saarbrücken.

Techn. University, Diss., 2004--Berlin.

The n-point Steiner problem in the Euclidean plane is to find a least length path network connecting n points. In this thesis we will demonstrate how to find a least length path network T connecting n points on a closed 2-dimensional Riemannian surface of constant curvature by determining a region in the covering space that is guaranteed to contain T. We will then provide an algorithm for solving the n-point Steiner problem on such a surface.

This thesis presents and develops a theory of self-reduction. This process is used to map instances of combinatorial optimisation problems onto smaller, more easily solvable instances in such a way that a solution of the former can be readily re-constructed, without loss of information or quality, from a solution of the latter. Self-reduction rules are surveyed for the Graph Colouring Problem, the Maximum Clique Problem, the Steiner Problem in Graphs, the Bin Packing Problem and the Set Covering Problem. This thesis introduces the problem of determining the maximum sequence of self-reductions on a given structure, and shows how the theory of confluence can be adapted from term re-writing to solve this problem by identifying rule sets for which all maximal reduction sequences are equivalent. Such confluence results are given for a number of reduction rules on problems on discrete systems. In contrast, NP-hardness results are also presented for some reduction rules. A probabilistic analysis of self-reductions on graphs is performed, showing that the expected number of self-reductions on a graph tends to zero as the order of the graph tends to infinity. An empirical study is performed comparing the performance of self-reduction, graph decomposition and direct methods of solving the Graph Colouring and Set Covering Problems. The results show that self-reduction is a potentially valuable, but sometimes erratic, method of finding exact solutions to combinatorial problems.

The subject of this monograph can be described as the local properties of geometric Steiner minimal trees in finite-dimensional normed spaces. A Steiner minimal tree of a finite set of points is a shortest connected set interconnecting the points. For a quick introduction to this topic and an overview of all the results presented in this work, see Chapter 1. The relevant mathematical background knowledge needed to understand the results and their proofs are collected in Chapter 2. In Chapter 3 we introduce the Fermat-Torricelli problem, which is that of finding a point that minimizes the sum of distances to a finite set of given points. We only develop that part of the theory of Fermat-Torricelli points that is needed in later chapters. Steiner minimal trees in finite-dimensional normed spaces are introduced in Chapter 4, where the local Steiner problem is given an exact formulation. In Chapter 5 we solve the local Steiner problem for all two-dimensional spaces, and generalize this solution to a certain class of higher-dimensional spaces (CL spaces). The twodimensional solution is then applied to many specific norms in Chapter 6. Chapter 7 contains an abstract solution valid in any dimension, based on the subdifferential calculus. This solution is applied to two specific high-dimensional spaces in Chapter 8. In Chapter 9 we introduce an alternative approach to bounding the maximum degree of Steiner minimal trees from above, based on the illumination problem from combinatorial convexity. Finally, in Chapter 10 we consider the related k-Steiner minimal trees, which are shortest Steiner trees in which the number of Steiner points is restricted to be at most k. / Das Thema dieser Habilitationsschrift kann als die lokalen Eigenschaften der geometrischen minimalen Steiner-Bäume in endlich-dimensionalen normierten Räumen beschrieben werden. Ein minimaler Steiner-Baum einer endlichen Punktmenge ist eine kürzeste zusammenhängende Menge die die Punktmenge verbindet. Kapitel 1 enthält eine kurze Einführung zu diesem Thema und einen Überblick über alle Ergebnisse dieser Arbeit. Die entsprechenden mathematischen Vorkenntnisse mit ihren Beweisen, die erforderlich sind die Ergebnisse zu verstehen, erscheinen in Kapitel 2. In Kapitel 3 führen wir das Fermat-Torricelli-Problem ein, das heißt, die Suche nach einem Punkt, der die Summe der Entfernungen der Punkte einer endlichen Punktmenge minimiert. Wir entwickeln nur den Teil der Theorie der Fermat-Torricelli-Punkte, der in späteren Kapiteln benötigt wird. Minimale Steiner-Bäume in endlich-dimensionalen normierten Räumen werden in Kapitel 4 eingeführt, und eine exakte Formulierung wird für das lokale Steiner-Problem gegeben. In Kapitel 5 lösen wir das lokale Steiner-Problem für alle zwei-dimensionalen Räume, und diese Lösung wird für eine bestimmte Klasse von höher-dimensionalen Räumen (den sog. CL-Räumen) verallgemeinert. Die zweidimensionale Lösung wird dann auf mehrere bestimmte Normen in Kapitel 6 angewandt. Kapitel 7 enthält eine abstrakte Lösung die in jeder Dimension gilt, die auf der Analysis von Subdifferentialen basiert. Diese Lösung wird auf zwei bestimmte höher-dimensionale Räume in Kapitel 8 angewandt. In Kapitel 9 führen wir einen alternativen Ansatz zur oberen Schranke des maximalen Grads eines minimalen Steiner-Baums ein, der auf dem Beleuchtungsproblem der kombinatorischen Konvexität basiert ist. Schließlich betrachten wir in Kapitel 10 die verwandten minimalen k-Steiner-Bäume. Diese sind die kürzesten Steiner-Bäume, in denen die Anzahl der Steiner-Punkte auf höchstens k beschränkt wird.