@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{Witek2014,
  author    = {Witek, Maximilian},
  title     = {Multiobjective Traveling Salesman Problems and Redundancy of Complete Sets},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-110740},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {The first part of this thesis deals with the approximability of the traveling salesman problem. This problem is defined on a complete graph with edge weights, and the task is to find a Hamiltonian cycle of minimum weight that visits each vertex exactly once. We study the most important multiobjective variants of this problem. In the multiobjective case, the edge weights are vectors of natural numbers with one component for each objective, and since weight vectors are typically incomparable, the optimal Hamiltonian cycle does not exist. Instead we consider the Pareto set, which consists of those Hamiltonian cycles that are not dominated by some other, strictly better Hamiltonian cycles. The central goal in multiobjective optimization and in the first part of this thesis in particular is the approximation of such Pareto sets. We first develop improved approximation algorithms for the two-objective metric traveling salesman problem on multigraphs and for related Hamiltonian path problems that are inspired by the single-objective Christofides' heuristic. We further show arguments indicating that our algorithms are difficult to improve. Furthermore we consider multiobjective maximization versions of the traveling salesman problem, where the task is to find Hamiltonian cycles with high weight in each objective. We generalize single-objective techniques to the multiobjective case, where we first compute a cycle cover with high weight and then remove an edge with low weight in each cycle. Since weight vectors are often incomparable, the choice of the edges of low weight is non-trivial. We develop a general lemma that solves this problem and enables us to generalize the single-objective maximization algorithms to the multiobjective case. We obtain improved, randomized approximation algorithms for the multiobjective maximization variants of the traveling salesman problem. We conclude the first part by developing deterministic algorithms for these problems. The second part of this thesis deals with redundancy properties of complete sets. We call a set autoreducible if for every input instance x we can efficiently compute some y that is different from x but that has the same membership to the set. If the set can be split into two equivalent parts, then it is called weakly mitotic, and if the splitting is obtained by an efficiently decidable separator set, then it is called mitotic. For different reducibility notions and complexity classes, we analyze how redundant its complete sets are. Previous research in this field concentrates on polynomial-time computable reducibility notions. The main contribution of this part of the thesis is a systematic study of the redundancy properties of complete sets for typical complexity classes and reducibility notions that are computable in logarithmic space. We use different techniques to show autoreducibility and mitoticity that depend on the size of the complexity class and the strength of the reducibility notion considered. For small complexity classes such as NL and P we use self-reducible, complete sets to show that all complete sets are autoreducible. For large complexity classes such as PSPACE and EXP we apply diagonalization methods to show that all complete sets are even mitotic. For intermediate complexity classes such as NP and the remaining levels of the polynomial-time hierarchy we establish autoreducibility of complete sets by locally checking computational transcripts. In many cases we can show autoreducibility of complete sets, while mitoticity is not known to hold. We conclude the second part by showing that in some cases, autoreducibility of complete sets at least implies weak mitoticity.},
  subject      = {Mehrkriterielle Optimierung},
  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}
}