Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

<p> This thesis is a contribution to the theory of Steiner quadruple systems. S-cyclic Steiner quadruple systems are defined and then as a main result it is shown that there exists exactly one S-cyclic Steiner quadruple system of order 20.</p> / Thesis / Master of Science (MSc)

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

The Steiner tree problem is a classical, well-studied, $\mathcal{NP}$-hard optimization problem. Here we are given an undirected graph $G=(V,E)$, a subset $R$ of $V$ of terminals, and non-negative costs $c_e$ for all edges $e$ in $E$. A feasible Steiner tree for a given instance is a tree $T$ in $G$ that spans all terminals in $R$. The goal is to compute a feasible Steiner tree of smallest cost. In this thesis we will focus on approximation algorithms for this problem: a $c$-approximation algorithm is an algorithm that returns a tree of cost at most $c$ times that of an optimum solution for any given input instance. <br /><br /> In a series of papers throughout the last decade, the approximation guarantee $c$ for the Steiner tree problem has been improved to the currently best known value of 1. 55 (Robins, Zelikovsky). Robins' and Zelikovsky's algorithm as well as most of its predecessors are greedy algorithms. <br /><br /> Apart from algorithmic improvements, there also has been substantial work on obtaining tight linear-programming relaxations for the Steiner tree problem. Many undirected and directed formulations have been proposed in the course of the last 25 years; their use, however, is to this point mostly restricted to the field of exact optimization. There are few examples of algorithms for the Steiner tree problem that make use of these LP relaxations. The best known such algorithm for general graphs is a 2-approximation (for the more general Steiner forest problem) due to Agrawal, Klein and Ravi. Their analysis is tight as the LP-relaxation used in their work is known to be weak: it has an IP/LP gap of approximately 2. <br /><br /> Most recent efforts to obtain algorithms for the Steiner tree problem that are based on LP-relaxations has focused on directed relaxations. In this thesis we present an undirected relaxation and show that the algorithm of Robins and Zelikovsky returns a Steiner tree whose cost is at most 1. 55 times its optimum solution value. In fact, we show that this algorithm can be viewed as a primal-dual algorithm. <br /><br/> The Steiner forest problem is a generalization of the Steiner tree problem. In the problem, instead of only one set of terminals, we are given more than one terminal set. An feasible Steiner forest is a forest that connects all terminals in the same terminal set for each terminal set. The goal is to find a minimum cost feasible Steiner forest. In this thesis, a new set of facet defining inequalities for the polyhedra of the Steiner forest is introduced.

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles.

The Steiner tree problem is a classical, well-studied, $\mathcal{NP}$-hard optimization problem. Here we are given an undirected graph $G=(V,E)$, a subset $R$ of $V$ of terminals, and non-negative costs $c_e$ for all edges $e$ in $E$. A feasible Steiner tree for a given instance is a tree $T$ in $G$ that spans all terminals in $R$. The goal is to compute a feasible Steiner tree of smallest cost. In this thesis we will focus on approximation algorithms for this problem: a $c$-approximation algorithm is an algorithm that returns a tree of cost at most $c$ times that of an optimum solution for any given input instance. <br /><br /> In a series of papers throughout the last decade, the approximation guarantee $c$ for the Steiner tree problem has been improved to the currently best known value of 1. 55 (Robins, Zelikovsky). Robins' and Zelikovsky's algorithm as well as most of its predecessors are greedy algorithms. <br /><br /> Apart from algorithmic improvements, there also has been substantial work on obtaining tight linear-programming relaxations for the Steiner tree problem. Many undirected and directed formulations have been proposed in the course of the last 25 years; their use, however, is to this point mostly restricted to the field of exact optimization. There are few examples of algorithms for the Steiner tree problem that make use of these LP relaxations. The best known such algorithm for general graphs is a 2-approximation (for the more general Steiner forest problem) due to Agrawal, Klein and Ravi. Their analysis is tight as the LP-relaxation used in their work is known to be weak: it has an IP/LP gap of approximately 2. <br /><br /> Most recent efforts to obtain algorithms for the Steiner tree problem that are based on LP-relaxations has focused on directed relaxations. In this thesis we present an undirected relaxation and show that the algorithm of Robins and Zelikovsky returns a Steiner tree whose cost is at most 1. 55 times its optimum solution value. In fact, we show that this algorithm can be viewed as a primal-dual algorithm. <br /><br/> The Steiner forest problem is a generalization of the Steiner tree problem. In the problem, instead of only one set of terminals, we are given more than one terminal set. An feasible Steiner forest is a forest that connects all terminals in the same terminal set for each terminal set. The goal is to find a minimum cost feasible Steiner forest. In this thesis, a new set of facet defining inequalities for the polyhedra of the Steiner forest is introduced.

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles.

Thesis (M.Ed. (Hons.))--Faculty of Education and Social Work, University of Sydney, 2005. / Bibliography: leaves 116-120.

Thesis (M.Ed. (Hons.))--Faculty of Education and Social Work, University of Sydney, 2005. / Bibliography: leaves 116-120.