Simultaneous Visualization of Clusterings

While there are a number of approaches for the visualization of hierarchical clusterings, as well as subsets in general, there exist only few attempts to visualize different clusterings simultaneously in the same drawing. In this thesis we lay the theoretical foundations for the simultaneous visualization of two or more clusterings. We establish a class-hierarchy that allows us to characterize families of clusterings depending on their embeddability in the plane and show how these classes relate to known combinatorial problems. For some of the classes it turns out to be an NP-complete problem to decide whether an instance belongs to the class or not. We develop and implement an integer linear program that allows us to find optimal embeddings for two classes and additionally provide a simple and fast heuristic that allows us to find a good, yet not always optimal solution. We finally experimentally evaluate our methods on randomly generated instances regarding the embeddability of clusterings and the quality of the heuristic. We conclude this work with a case study in which we apply our methods to two examples based on real-world data.