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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.
In many cases, problems, data, or information can be modeled as graphs. Graphs can be used as a tool for modeling in any case where connections between distinguishable objects occur. Any graph consists of a set of objects, called vertices, and a set of connections, called edges, such that any edge connects a pair of vertices. For example, a social network can be modeled by a graph by
transforming the users of the network into vertices and friendship relations between users into edges. Also physical networks like computer networks or transportation networks, for example, the metro network of a city, can be seen as graphs.
For making graphs and, thereby, the data that is modeled, well-understandable for users, we need a visualization. Graph drawing deals with algorithms for visualizing graphs. In this thesis, especially the use of crossings and curves is investigated for graph drawing problems under additional constraints. The constraints that occur in the problems investigated in this thesis especially restrict the positions of (a part of) the vertices; this is done either as a hard constraint or as an optimization criterion.
Given points in the plane, connect them using minimum ink. Though the task seems simple, it turns out to be very time consuming. In fact, scientists believe that computers cannot efficiently solve it. So, do we have to resign? This book examines such NP-hard network-design problems, from connectivity problems in graphs to polygonal drawing problems on the plane. First, we observe why it is so hard to optimally solve these problems. Then, we go over to attack them anyway. We develop fast algorithms that find approximate solutions that are very close to the optimal ones. Hence, connecting points with slightly more ink is not hard.
The Software Defined Networking (SDN) paradigm offers network operators numerous improvements in terms of flexibility, scalability, as well as cost efficiency and vendor independence. However, in order to maximize the benefit from these features, several new challenges in areas such as management and orchestration need to be addressed. This dissertation makes contributions towards three key topics from these areas.
Firstly, we design, implement, and evaluate two multi-objective heuristics for the SDN controller placement problem. Secondly, we develop and apply mechanisms for automated decision making based on the Pareto frontiers that are returned by the multi-objective optimizers. Finally, we investigate and quantify the performance benefits for the SDN control plane that can be achieved by integrating information from external entities such as Network Management Systems (NMSs) into the control loop. Our evaluation results demonstrate the impact of optimizing various parameters of softwarized networks at different levels and are used to derive guidelines for an efficient operation.
In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974.
We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree.
Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs.
Besides that, we study spectra of zero-divisor graphs over finite commutative rings.
Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.