@phdthesis{Fleszar2018, author = {Fleszar, Krzysztof}, title = {Network-Design Problems in Graphs and on the Plane}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-076-4 (Print)}, doi = {10.25972/WUP-978-3-95826-077-1}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-154904}, school = {W{\"u}rzburg University Press}, pages = {xi, 204}, year = {2018}, abstract = {A network design problem defines an infinite set whose elements, called instances, describe relationships and network constraints. It asks for an algorithm that, given an instance of this set, designs a network that respects the given constraints and at the same time optimizes some given criterion. In my thesis, I develop algorithms whose solutions are optimum or close to an optimum value within some guaranteed bound. I also examine the computational complexity of these problems. Problems from two vast areas are considered: graphs and the Euclidean plane. In the Maximum Edge Disjoint Paths problem, we are given a graph and a subset of vertex pairs that are called terminal pairs. We are asked for a set of paths where the endpoints of each path form a terminal pair. The constraint is that any two paths share at most one inner vertex. The optimization criterion is to maximize the cardinality of the set. In the hard-capacitated k-Facility Location problem, we are given an integer k and a complete graph where the distances obey a given metric and where each node has two numerical values: a capacity and an opening cost. We are asked for a subset of k nodes, called facilities, and an assignment of all the nodes, called clients, to the facilities. The constraint is that the number of clients assigned to a facility cannot exceed the facility's capacity value. The optimization criterion is to minimize the total cost which consists of the total opening cost of the facilities and the total distance between the clients and the facilities they are assigned to. In the Stabbing problem, we are given a set of axis-aligned rectangles in the plane. We are asked for a set of horizontal line segments such that, for every rectangle, there is a line segment crossing its left and right edge. The optimization criterion is to minimize the total length of the line segments. In the k-Colored Non-Crossing Euclidean Steiner Forest problem, we are given an integer k and a finite set of points in the plane where each point has one of k colors. For every color, we are asked for a drawing that connects all the points of the same color. The constraint is that drawings of different colors are not allowed to cross each other. The optimization criterion is to minimize the total length of the drawings. In the Minimum Rectilinear Polygon for Given Angle Sequence problem, we are given an angle sequence of left (+90°) turns and right (-90°) turns. We are asked for an axis-parallel simple polygon where the angles of the vertices yield the given sequence when walking around the polygon in counter-clockwise manner. The optimization criteria considered are to minimize the perimeter, the area, and the size of the axis-parallel bounding box of the polygon.}, subject = {Euklidische Ebene}, language = {en} } @phdthesis{Reitwiessner2011, author = {Reitwießner, Christian}, title = {Multiobjective Optimization and Language Equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-70146}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {Praktische Optimierungsprobleme beinhalten oft mehrere gleichberechtigte, sich jedoch widersprechende Kriterien. Beispielsweise will man bei einer Reise zugleich m{\"o}glichst schnell ankommen, sie soll aber auch nicht zu teuer sein. Im ersten Teil dieser Arbeit wird die algorithmische Beherrschbarkeit solcher mehrkriterieller Optimierungsprobleme behandelt. Es werden zun{\"a}chst verschiedene L{\"o}sungsbegriffe diskutiert und auf ihre Schwierigkeit hin verglichen. Interessanterweise stellt sich heraus, dass diese Begriffe f{\"u}r ein einkriterielles Problem stets gleich schwer sind, sie sich ab zwei Kriterien allerdings stark unterscheiden k{\"o}nen (außer es gilt P = NP). In diesem Zusammenhang wird auch die Beziehung zwischen Such- und Entscheidungsproblemen im Allgemeinen untersucht. Schließlich werden neue und verbesserte Approximationsalgorithmen f{\"u}r verschieden Varianten des Problems des Handlungsreisenden gefunden. Dabei wird mit Mitteln der Diskrepanztheorie eine Technik entwickelt, die ein grundlegendes Hindernis der Mehrkriteriellen Optimierung aus dem Weg schafft: Gegebene L{\"o}sungen so zu kombinieren, dass die neue L{\"o}sung in allen Kriterien m{\"o}glichst ausgewogen ist und gleichzeitig die Struktur der L{\"o}sungen nicht zu stark zerst{\"o}rt wird. Der zweite Teil der Arbeit widmet sich verschiedenen Aspekten von Gleichungssystemen f{\"u}r (formale) Sprachen. Einerseits werden konjunktive und Boolesche Grammatiken untersucht. Diese sind Erweiterungen der kontextfreien Grammatiken um explizite Durchschnitts- und Komplementoperationen. Es wird unter anderem gezeigt, dass man bei konjunktiven Grammatiken die Vereinigungsoperation stark einschr{\"a}nken kann, ohne dabei die erzeugte Sprache zu {\"a}ndern. Außerdem werden bestimmte Schaltkreise untersucht, deren Gatter keine Wahrheitswerte sondern Mengen von Zahlen berechnen. F{\"u}r diese Schaltkreise wird das {\"A}quivalenzproblem betrachtet, also die Frage ob zwei gegebene Schaltkreise die gleiche Menge berechnen oder nicht. Es stellt sich heraus, dass, abh{\"a}ngig von den erlaubten Gattertypen, die Komplexit{\"a}t des {\"A}quivalenzproblems stark variiert und f{\"u}r verschiedene Komplexit{\"a}tsklassen vollst{\"a}ndig ist, also als (parametrisierter) Vertreter f{\"u}r diese Klassen stehen kann.}, subject = {Mehrkriterielle Optimierung}, language = {en} } @phdthesis{Spoerhase2009, author = {Spoerhase, Joachim}, title = {Competitive and Voting Location}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-52978}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2009}, abstract = {We consider competitive location problems where two competing providers place their facilities sequentially and users can decide between the competitors. We assume that both competitors act non-cooperatively and aim at maximizing their own benefits. We investigate the complexity and approximability of such problems on graphs, in particular on simple graph classes such as trees and paths. We also develop fast algorithms for single competitive location problems where each provider places a single facilty. Voting location, in contrast, aims at identifying locations that meet social criteria. The provider wants to satisfy the users (customers) of the facility to be opened. In general, there is no location that is favored by all users. Therefore, a satisfactory compromise has to be found. To this end, criteria arising from voting theory are considered. The solution of the location problem is understood as the winner of a virtual election among the users of the facilities, in which the potential locations play the role of the candidates and the users represent the voters. Competitive and voting location problems turn out to be closely related.}, subject = {Standortproblem}, language = {en} }