Martin Nöllenburg

Alexander Dobler, Stephen Kobourov, Debajyoti Mondal et al.

We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a) restricting the curves to be lines or line segments, (b) allowing two curves to cross if they do not share an element, or not; and (c) allowing two curves to overlap or not. We show $\exists\mathbb{R}$-hardness for six of the eight resulting decision problem variants and describe polynomial-time algorithms in some restricted settings. Lastly, we briefly touch on what happens if we allow the lines of the represented hyperedges to have bends - to this we generalize a counterexample to a long-standing result that was sometimes assumed to be correct.

Martin Nöllenburg, Christian Pirker, Anna Rapberger et al.

The visualization of argumentation frameworks (AFs) is crucial for enabling a wide applicability of argumentative tools. However, their visualization is often considered only as an accompanying part of tools for computing semantics and standard graphical representations are used. We introduce a new visualization technique that draws an AF, together with an extension (as part of the input), as a 3-layer graph layout. Our technique supports the user to more easily explore the visualized AF, better understand extensions, and verify algorithms for computing semantics. To optimize the visual clarity and aesthetics of this layout, we propose to minimize edge crossings in our 3-layer drawing. We do so by an exact ILP-based approach, but also propose a fast heuristic pipeline. Via a quantitative evaluation, we show that the heuristic is feasible even for large instances, while producing at most twice as many crossings as an optimal drawing in most cases.

Panos Giannopoulos, Miriam Goetze, Grzegorz Gutowski et al.

Drawing graphs with the minimum number of crossings is a classical problem that has been studied extensively. Many restricted versions of the problem have been considered. For example, bipartite graphs can be drawn such that the two sets in the bipartition of the vertex set are mapped to two parallel lines, and the edges are drawn as straight-line segments. In this setting, the number of crossings depends only on the ordering of the vertices on the two lines. Two natural variants of the problem have been studied. In the one-sided case, the order of the vertices on one of the two lines is given and fixed; in the two-sided case, no order is given. Both cases are important yet NP-hard subproblems in the so-called Sugiyama framework for drawing layered graphs with few crossings. For the one-sided case, Eades and Wormald [Algorithmica 1994] introduced a median heuristic and showed that it has an approximation ratio of 3. In recent years, researchers have focused on a local version of crossing minimization, where the aim is to minimize the maximum number of crossings per edge instead of the total number of crossings. Kobayashi, Okada, and Wolff [SoCG 2025] investigated the complexity of local crossing minimization parameterized by the natural parameter. They conjectured that one-sided local crossing minimization is NP-hard. In this work, we confirm their conjecture by showing that the problem is NP-hard even for forests of high-degree stars. In fact, more strongly, the reduction yields a tight lower bound, which excludes the existence of subexponential-time algorithms assuming the Exponential-Time Hypothesis. In contrast, we present a quadratic-time algorithm for the special case of forests of stars of maximum degree 2. Finally, we provide a median heuristic with a carefully designed tie-breaking scheme and prove that it has an approximation ratio of 3 in the local setting.

Hsiang-Yun Wu, Martin Nöllenburg, Ivan Viola

The concept of multilayer networks has become recently integrated into complex systems modeling since it encapsulates a very general concept of complex relationships. Biological pathways are an example of complex real-world networks, where vertices represent biological entities, and edges indicate the underlying connectivity. For this reason, using multilayer networks to model biological knowledge allows us to formally cover essential properties and theories in the field, which also raises challenges in visualization. This is because, in the early days of pathway visualization research, only restricted types of graphs, such as simple graphs, clustered graphs, and others were adopted. In this paper, we revisit a heterogeneous definition of biological networks and aim to provide an overview to see the gaps between data modeling and visual representation. The contribution will, therefore, lie in providing guidelines and challenges of using multilayer networks as a unified data structure for the biological pathway visualization.

Sujoy Bhore, Robert Ganian, Guangping Li et al.

Point feature labeling is a classical problem in cartography and GIS that has been extensively studied for geospatial point data. At the same time, word clouds are a popular visualization tool to show the most important words in text data which has also been extended to visualize geospatial data (Buchin et al. PacificVis 2016). In this paper, we study a hybrid visualization, which combines aspects of word clouds and point labeling. In the considered setting, the input data consists of a set of points grouped into categories and our aim is to place multiple disjoint and axis-aligned rectangles, each representing a category, such that they cover points of (mostly) the same category under some natural quality constraints. In our visualization, we then place category names inside the computed rectangles to produce a labeling of the covered points which summarizes the predominant categories globally (in a word-cloud-like fashion) while locally avoiding excessive misrepresentation of points (i.e., retaining the precision of point labeling). We show that computing a minimum set of such rectangles is NP-hard. Hence, we turn our attention to developing heuristics and exact SAT models to compute our visualizations. We evaluate our algorithms quantitatively, measuring running time and quality of the produced solutions, on several artificial and real-world data sets. Our experiments show that the heuristics produce solutions of comparable quality to the SAT models while running much faster.

Markus Wallinger, Ben Jacobsen, Stephen Kobourov et al.

Set systems are used to model data that naturally arises in many contexts: social networks have communities, musicians have genres, and patients have symptoms. Visualizations that accurately reflect the information in the underlying set system make it possible to identify the set elements, the sets themselves, and the relationships between the sets. In static contexts, such as print media or infographics, it is necessary to capture this information without the help of interactions. With this in mind, we consider three different systems for medium-sized set data, LineSets, EulerView, and MetroSets, and report the results of a controlled human-subjects experiment comparing their effectiveness. Specifically, we evaluate the performance, in terms of time and error, on tasks that cover the spectrum of static set-based tasks. We also collect and analyze qualitative data about the three different visualization systems. Our results include statistically significant differences, suggesting that MetroSets performs and scales better.

Ben Jacobsen, Markus Wallinger, Stephen Kobourov et al.

We propose MetroSets, a new, flexible online tool for visualizing set systems using the metro map metaphor. We model a given set system as a hypergraph $\mathcal{H} = (V, \mathcal{S})$, consisting of a set $V$ of vertices and a set $\mathcal{S}$, which contains subsets of $V$ called hyperedges. Our system then computes a metro map representation of $\mathcal{H}$, where each hyperedge $E$ in $\mathcal{S}$ corresponds to a metro line and each vertex corresponds to a metro station. Vertices that appear in two or more hyperedges are drawn as interchanges in the metro map, connecting the different sets. MetroSets is based on a modular 4-step pipeline which constructs and optimizes a path-based hypergraph support, which is then drawn and schematized using metro map layout algorithms. We propose and implement multiple algorithms for each step of the MetroSet pipeline and provide a functional prototype with easy-to-use preset configurations. Furthermore, using several real-world datasets, we perform an extensive quantitative evaluation of the impact of different pipeline stages on desirable properties of the generated maps, such as octolinearity, monotonicity, and edge uniformity.

Fabian Klute, Guangping Li, Raphael Löffler et al.

Label placement in maps is a very challenging task that is critical for the overall map quality. Most previous work focused on designing and implementing fully automatic solutions, but the resulting visual and aesthetic quality has not reached the same level of sophistication that skilled human cartographers achieve. We investigate a different strategy that combines the strengths of humans and algorithms. In our proposed method, first an initial labeling is computed that has many well-placed labels but is not claiming to be perfect. Instead it serves as a starting point for an expert user who can then interactively and locally modify the labeling where necessary. In an iterative human-in-the-loop process alternating between user modifications and local algorithmic updates and refinements the labeling can be tuned to the user's needs. We demonstrate our approach by performing different possible modification steps in a sample workflow with a prototypical interactive labeling editor. Further, we report computational performance results from a simulation experiment in QGIS, which investigates the differences between exact and heuristic algorithms for semi-automatic map labeling. To that end, we compare several alternatives for recomputing the labeling after local modifications and updates, as a major ingredient for an interactive labeling editor.

Yifan Hu, Martin Nöllenburg

This is the arXiv index for the electronic proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016), which was held in Athens, Greece, September 19-21 2016. It contains the peer-reviewed and revised accepted papers with an optional appendix.