arXiv subjects

Martin Nöllenburg

Publications and source records attributed to Martin Nöllenburg.

At least 55 records · Page 3Linked to original sources

arXiv2023

Splitting Plane Graphs to Outerplanarity

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is NP-hard. Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove NP-completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Finally, we provide upper and lower bounds for certain families of maximal planar graphs.

cs.CG↗

arXiv2022

Minimum Link Fencing

We study a variant of the geometric multicut problem, where we are given a set $\mathcal{P}$ of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where $\mathcal{P}$ contains a polygon $Q$ that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an $O(n \log n)$-time algorithm for the case that the convex hull of $\mathcal{P} \setminus \{Q\}$ does not intersect $Q$.

cs.CG↗

arXiv2022

Planarizing Graphs and their Drawings by Vertex Splitting

The splitting number of a graph $G=(V,E)$ is the minimum number of vertex splits required to turn $G$ into a planar graph, where a vertex split removes a vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the edges formerly incident to $v$ among its two split copies $v_1, v_2$. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in $\mathbb R^2$, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform fixed-parameter tractable (FPT) algorithm for the splitting number problem (without drawings). Then we show the NP-completeness of the splitting number problem for graph drawings, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.

cs.CG↗

arXiv2022

On the Complexity of the Storyplan Problem

Motivated by dynamic graph visualization, we study the problem of representing a graph $G$ in the form of a \emph{storyplan}, that is, a sequence of frames with the following properties. Each frame is a planar drawing of the subgraph of $G$ induced by a suitably defined subset of its vertices. Between two consecutive frames, a new vertex appears while some other vertices may disappear, namely those whose incident edges have already been drawn in at least one frame. In a storyplan, each vertex appears and disappears exactly once. For a vertex (edge) visible in a sequence of consecutive frames, the point (curve) representing it does not change throughout the sequence. Note that the order in which the vertices of $G$ appear in the sequence of frames is a total order. In the \textsc{StoryPlan} problem, we are given a graph and we want to decide whether there exists a total order of its vertices for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of $G$. Also, we prove that partial $3$-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete in the case in which the total order of the vertices is given as part of the input and we have to choose how to draw the frames.

cs.CC↗

arXiv2022

Shape-Guided Mixed Metro Map Layout

Metro or transit maps, are schematic representations of transit networks to facilitate effective route-finding. These maps are often advertised on a web page or pamphlet highlighting routes from source to destination stations. To visually support such route-finding, designers often distort the layout by embedding symbolic shapes (e.g., circular routes) in order to guide readers' attention (e.g., Moscow map and Japan railway map). However, manually producing such maps is labor-intensive and the effect of shapes remains unclear. In this paper, we propose an approach to generalize such mixed metro maps that take user-defined shapes as an input. In this mixed design, lines that are used to approximate the shapes are arranged symbolically, while the remaining lines follow classical layout convention. A three-step algorithm, including (1) detecting and selecting routes for shape approximation, (2) shape and layout deformation, and (3) aligning lines on a grid, is integrated to guarantee good visual quality. Our contribution lies in the definition of the mixed metro map problem and the formulation of design criteria so that the problem can be resolved systematically using the optimization paradigm. Finally, we evaluate the performance of our approach and perform a user study to test if the embedded shapes are recognizable or reduce the map quality.

cs.GR↗

arXiv2022

On Computing Optimal Linear Diagrams

Linear diagrams are an effective way to visualize set-based data by representing elements as columns and sets as rows with one or more horizontal line segments, whose vertical overlaps with other rows indicate set intersections and their contained elements. The efficacy of linear diagrams heavily depends on having few line segments. The underlying minimization problem has already been explored heuristically, but its computational complexity has yet to be classified. In this paper, we show that minimizing line segments in linear diagrams is equivalent to a well-studied NP-hard problem, and extend the NP-hardness to a restricted setting. We develop new algorithms for computing linear diagrams with minimum number of line segments that build on a traveling salesperson (TSP) formulation and allow constraints on the element orders, namely, forcing two sets to be drawn as single line segments, giving weights to sets, and allowing hierarchical constraints via PQ-trees. We conduct an experimental evaluation and compare previous algorithms for minimizing line segments with our TSP formulation, showing that a state-of-the art TSP-solver can solve all considered instances optimally, most of them within few milliseconds.

cs.CG↗

arXiv2022

MosaicSets: Embedding Set Systems into Grid Graphs

Visualizing sets of elements and their relations is an important research area in information visualization. In this paper, we present MosaicSets: a novel approach to create Euler-like diagrams from non-spatial set systems such that each element occupies one cell of a regular hexagonal or square grid. The main challenge is to find an assignment of the elements to the grid cells such that each set constitutes a contiguous region. As use case, we consider the research groups of a university faculty as elements, and the departments and joint research projects as sets. We aim at finding a suitable mapping between the research groups and the grid cells such that the department structure forms a base map layout. Our objectives are to optimize both the compactness of the entirety of all cells and of each set by itself. We show that computing the mapping is NP-hard. However, using integer linear programming we can solve real-world instances optimally within a few seconds. Moreover, we propose a relaxation of the contiguity requirement to visualize otherwise non-embeddable set systems. We present and discuss different rendering styles for the set overlays. Based on a case study with real-world data, our evaluation comprises quantitative measures as well as expert interviews.

cs.GR↗

arXiv2022

A Survey on Computing Schematic Network Maps: The Challenge to Interactivity

Schematic maps are in daily use to show the connectivity of subway systems and to facilitate travellers to plan their journeys effectively. This study surveys up-to-date algorithmic approaches in order to give an overview of the state of the art in schematic network mapping. The study investigates the hypothesis that the choice of algorithmic approach is often guided by the requirements of the mapping application. For example, an algorithm that computes globally optimal solutions for schematic maps is capable of producing results for printing, while it is not suitable for computing instant layouts due to its long running time. Our analysis and discussion, therefore, focus on the computational complexity of the problem formulation and the running times of the schematic map algorithms, including algorithmic network layout techniques and station labeling techniques. The correlation between problem complexity and running time is then visually depicted using scatter plot diagrams. Moreover, since metro maps are common metaphors for data visualization, we also investigate online tools and application domains using metro map representations for analytics purposes, and finally summarize the potential future opportunities for schematic maps.

physics.soc-ph↗

arXiv2022

Geometric Planar Networks on Bichromatic Points

We study three classical graph problems - Hamiltonian path, minimum spanning tree, and minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP-hard. In this work, we consider these problems for collinear points. We show that almost all of these problems can be solved in linear time in this setting.

cs.CG↗

arXiv2022

Balanced Independent and Dominating Sets on Colored Interval Graphs

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) and $f$-Balanced Dominating Set ($f$-BDS). Let $G=(V,E)$ be a vertex-colored interval graph with a color assignment function $γ\colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset of vertices $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ vertices from each color class. In the $f$-BIS and $f$-BDS problems, the objective is to compute an independent set or a dominating set that is $f$-balanced. We show that both problems are NP-complete even on proper interval graphs. For the $f$-BIS problem, we design two FPT algorithms, one parameterized by $(f,k)$ for interval graphs and the other parameterized by the vertex cover number for general graphs. Moreover, for an optimization variant of BIS on interval graphs, we show that a simple greedy approach achieves an approximation ratio of $2$.

cs.DS↗

arXiv2022

An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling

Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set ((MIS)) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types -- (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of $Ω(Δ)$ update time (where $Δ$ is the maximum degree in the graph) for \emph{vertex updates} presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes with approximation factors between 2 and 4 and corresponding running-time trade-offs. We have implemented our algorithms and reported the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.

cs.CG↗

arXiv2021

Untangling Circular Drawings: Algorithms and Complexity

We consider the problem of untangling a given (non-planar) straight-line circular drawing $δ_G$ of an outerplanar graph $G=(V, E)$ into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph $G$, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift$(δ_G)$ as the minimum number of vertices that are required to be shifted in order to resolve all crossings of $δ_G$. We show that the problem Circular Untangling, asking whether shift$(δ_G) \le K$ for a given integer $K$, is NP-complete. For $n$-vertex outerplanar graphs, we obtain a tight upper bound of shift$(δ_G) \le n - \lfloor\sqrt{n-2}\rfloor -2$. Based on these results we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case, we provide a tight upper bound shift$(δ_G) \le \lfloor \frac{n}{2} \rfloor-1$ and present a constructive polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.

cs.CG↗

arXiv2021

Graph Models for Biological Pathway Visualization: State of the Art and Future Challenges

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.

cs.HC↗

arXiv2021

Worbel: Aggregating Point Labels into Word Clouds

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.

cs.CG↗

arXiv2021

On the Upward Book Thickness Problem: Combinatorial and Complexity Results

A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are $at$-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness $k$ is NP-hard for any fixed $k \ge 3$. We show that the problem, for any $k \ge 5$, remains NP-hard for graphs whose domination number is $O(k)$, but it is FPT in the vertex cover number.

cs.DM↗

arXiv2021

Unit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs

A unit disk intersection representation (UDR) of a graph $G$ represents each vertex of $G$ as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in $G$. A UDR with interior-disjoint disks is called a unit disk contact representation (UDC). We prove that it is NP-hard to decide if an outerplanar graph or an embedded tree admits a UDR. We further provide a linear-time decidable characterization of caterpillar graphs that admit a UDR. Finally we show that it can be decided in linear time if a lobster graph admits a weak UDC, which permits intersections between disks of non-adjacent vertices.

cs.CG↗

arXiv2021

Layered Area-Proportional Rectangle Contact Representations

We investigate two optimization problems on area-proportional rectangle contact representations for layered, embedded planar graphs. The vertices are represented as interior-disjoint unit-height rectangles of prescribed widths, grouped in one row per layer, and each edge is ideally realized as a rectangle contact of positive length. Such rectangle contact representations find applications in semantic word or tag cloud visualizations, where a collection of words is displayed such that pairs of semantically related words are close to each other. In this paper, we want to maximize the number of realized rectangle contacts or minimize the overall area of the rectangle contact representation, while avoiding any false adjacencies. We present a network flow model for area minimization, a linear-time algorithm for contact maximization of two-layer graphs, and an ILP model for maximizing contacts of $k$-layer graphs.

cs.CG↗

arXiv2021

Edge-Path Bundling: A Less Ambiguous Edge Bundling Approach

Edge bundling techniques cluster edges with similar attributes (i.e. similarity in direction and proximity) together to reduce the visual clutter. All edge bundling techniques to date implicitly or explicitly cluster groups of individual edges, or parts of them, together based on these attributes. These clusters can result in ambiguous connections that do not exist in the data. Confluent drawings of networks do not have these ambiguities, but require the layout to be computed as part of the bundling process. We devise a new bundling method, Edge-Path bundling, to simplify edge clutter while greatly reducing ambiguities compared to previous bundling techniques. Edge-Path bundling takes a layout as input and clusters each edge along a weighted, shortest path to limit its deviation from a straight line. Edge-Path bundling does not incur independent edge ambiguities typically seen in all edge bundling methods, and the level of bundling can be tuned through shortest path distances, Euclidean distances, and combinations of the two. Also, directed edge bundling naturally emerges from the model. Through metric evaluations, we demonstrate the advantages of Edge-Path bundling over other techniques.

cs.GR↗