@phdthesis{Katzmann2023,
  author    = {Katzmann, Maximilian},
  title     = {About the analysis of algorithms on networks with underlying hyperbolic geometry},
  doi       = {10.25932/publishup-58296},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-582965},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xi, 191},
  year      = {2023},
  abstract  = {Many complex systems that we encounter in the world can be formalized using networks. Consequently, they have been in the focus of computer science for decades, where algorithms are developed to understand and utilize these systems. Surprisingly, our theoretical understanding of these algorithms and their behavior in practice often diverge significantly. In fact, they tend to perform much better on real-world networks than one would expect when considering the theoretical worst-case bounds. One way of capturing this discrepancy is the average-case analysis, where the idea is to acknowledge the differences between practical and worst-case instances by focusing on networks whose properties match those of real graphs. Recent observations indicate that good representations of real-world networks are obtained by assuming that a network has an underlying hyperbolic geometry. In this thesis, we demonstrate that the connection between networks and hyperbolic space can be utilized as a powerful tool for average-case analysis. To this end, we first introduce strongly hyperbolic unit disk graphs and identify the famous hyperbolic random graph model as a special case of them. We then consider four problems where recent empirical results highlight a gap between theory and practice and use hyperbolic graph models to explain these phenomena theoretically. First, we develop a routing scheme, used to forward information in a network, and analyze its efficiency on strongly hyperbolic unit disk graphs. For the special case of hyperbolic random graphs, our algorithm beats existing performance lower bounds. Afterwards, we use the hyperbolic random graph model to theoretically explain empirical observations about the performance of the bidirectional breadth-first search. Finally, we develop algorithms for computing optimal and nearly optimal vertex covers (problems known to be NP-hard) and show that, on hyperbolic random graphs, they run in polynomial and quasi-linear time, respectively. Our theoretical analyses reveal interesting properties of hyperbolic random graphs and our empirical studies present evidence that these properties, as well as our algorithmic improvements translate back into practice.},
  language  = {en}
}