Graph Drawing Problem Research Articles

Solving the incremental graph drawing problem by multiple neighborhood solution-based tabu search algorithm

With the advent of the era of big data, more and more complex networks and knowledge based systems require data visualization for analysis and presentation, where graphs have become a standard representation model. Graph drawing addresses the problems of constructing geometric representations of graphs to make them easy to analyze. In this paper we aim to reduce the edge crossing number in the context of the incremental graph drawing problem (IGDP), in which we want to preserve the layout of a graph over successive drawings to present complex networks or knowledge-based systems. Specifically, we propose a metaheuristic algorithm called multiple neighborhood solution-based tabu search (MNSB-TS) by combining a solution-based tabu strategy and a multiple neighborhood structure for solving the incremental graph drawing problem. Our MNSB-TS approach introduces four neighborhood operators to accompany the solution-based tabu strategy for determining the tabu status of each neighbor solution. Extensive computational experiments on public benchmark instances demonstrate that the proposed MNSB-TS is highly competitive in comparison with the state-of-the-art heuristics and the exact optimization solver Gurobi. Key components of the approach are analyzed to evaluate their impact on algorithm performance and learn which search mechanisms are better suited for incremental graph drawing.

Crossing edge minimization in radial outerplanar layered graphs using segment paths

Crossing minimization is a natural problem in graph drawing, which consists of drawing it in the plane with a minimum number of crossings on the edges. We present an algorithm to minimize the crossing edges in radial layered graphs. The algorithm determines the optimal location of nodes in their corresponding circle layer and consists of two phases. In the first phase, a counterclockwise rotation of concentric circles is performed; in the next phase, edges are converted into segment paths. Unlike other algorithms with circular drawings, our algorithm does not require the insertion or removal of nodes and edges; only rotation and path construction operations are utilized. Two versions of the algorithm were created; the exhaustive version tests all the possible paths, while the random version tests a few number of paths. The goal is to minimize the crossing edges to obtain a clear visualization using three criteria: rotation, crossing edge detection and segment path construction. The algorithm has been successfully implemented in 30 instances of graphs with 20, 35 and 50 layers, where crossings are minimized in 86–93%, 93–96% and 95–97%, respectively.

Bitonic st-Orderings for Upward Planar Graphs: Splits and Bends in the Variable Embedding Scenario

Bitonic st-orderings for st-planar graphs were introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge in polynomial area. For an st-planar graph that does not admit a bitonic st-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input st-planar graph. The best-known lower bound on the number of required splits of an st-planar graph with n vertices is n-3. However, it is possible to compute a bitonic st-ordering without any split for the st-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings in polynomial area, the former translates into n-3 bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an st-planar graph that needs at least n-5 splits in both orientations. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.

The Complexity of Drawing a Graph in a Polygonal Region

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This establishes a wider context for the NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to have bounded bit complexity for the special case of a convex region, or for drawing a path in any polygonal region. In addition, we prove a Mnëv-type universality result - loosely speaking, that the solution spaces of instances of our graph drawing problem are equivalent, in a topological and algebraic sense, to bounded algebraic varieties.

Comments on: Tabu search tutorial. A Graph Drawing Application

As suggested by its title, the aim of this paper is to discuss how tabu search can be successfully applied to find high-quality solutions for a wide family of NP-hard optimization problems arising in a huge number of applied mathematics areas. The authors show that a graph drawing problem application is one of those excellent examples, given that since ever geometric representations play an important role in many real-world scenarios and are an important applied branch of graph theory.

A variable depth neighborhood search algorithm for the Min–Max Arc Crossing Problem

The Min–Max Arc Crossing Problem (MMACP) aims to minimize the maximum number of crossings over all arcs in a layered graph. It is a recently proposed variant of the well-known graph drawing problem where the total number of arc crossings is minimized. MMACP is of interest in VLSI design and graph visualization, especially when dealing with very large graphs. In this paper, we present a variable depth neighborhood search algorithm for solving MMACP. Our algorithm exploits good data structures, efficient neighborhood search schemes, and a carefully defined ejection chain scheme. Effectively embedding these basic structures within an iterated local search framework, our algorithm produced superior outcomes. In particular, the algorithm produced new solutions improving the best-known results for 154 out of the 301 benchmark instances available in the public domain. As a result, our paper enhances the state-of-the-art knowledge in graph drawing and arc crossing minimization. We also computationally analyzed the roles and effectiveness of various crucial features of our algorithm which helped us make informed recommendations for default parameter settings.

On the Complexity of Some Geometric Problems With Fixed Parameters

The following graph-drawing problems are known to be complete for the existential theory of the reals (${\exists \mathbb{R}}$-complete) as long as the parameter $k$ is unbounded. Do they remain ${\exists \mathbb{R}}$-complete for a fixed value $k$? Do $k$ graphs on a shared vertex set have a simultaneous geometric embedding? Is $G$ a segment intersection graph, where $G$ has maximum degree at most $k$? Given a graph $G$ with a rotation system and maximum degree at most $k$, does $G$ have a straight-line drawing which realizes the rotation system? We show that these, and some related, problems remain ${\exists \mathbb{R}}$-complete for constant $k$, where $k$ is in the double or triple digits. To obtain these results we establish a new variant of Mnëv's universality theorem, in which the gadgets are placed so as to interact minimally; this variant leads to fixed values for $k$, where the traditional variants of the universality theorem require unbounded values of $k$.

Drawing graphs on few lines and few planes

We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many relations to other challenging graph-drawing problems such as small-area or small-volume drawings,layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on asymptotically fewer lines than in the plane.

RAC drawings in subcubic area

In this paper, we study tradeoffs between curve complexity and area of Right Angle Crossing drawings (RAC drawings), which is a challenging theoretical problem in graph drawing. Given a graph with n vertices and m edges, we provide a RAC drawing algorithm with curve complexity 6 and area O(n2.75), which takes time O(n+m). Our algorithm improves the previous upper bound O(n3), by Di Giacomo et al., on the area of RAC drawings.

Tabu search for min-max edge crossing in graphs

Graph drawing is a key issue in the field of data analysis, given the ever-growing amount of information available today that require the use of automatic tools to represent it. Graph Drawing Problems (GDP) are hard combinatorial problems whose applications have been widely relevant in fields such as social network analysis and project management. While classically in GDPs the main aesthetic concern is related to the minimization of the total sum of crossing in the graph (min-sum), in this paper we focus on a particular variant of the problem, the Min-Max GDP, consisting in the minimization of the maximum crossing among all egdes. Recently proposed in scientific literature, the Min-Max GDP is a challenging variant of the original min-sum GDP arising in the optimization of VLSI circuits and the design of interactive graph drawing tools. We propose a heuristic algorithm based on the tabu search methodology to obtain high-quality solutions. Extensive experimentation on an established benchmark set with both previous heuristics and optimal solutions shows that our method is able to obtain excellent solutions in short computation time.

Ordered Level Planarity and Its Relationship to Geodesic Planarity, Bi-Monotonicity, and Variations of Level Planarity

We introduce and study the problem Ordered Level Planarity, which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y -monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge e is realized as a polygonal path p composed of line segments with two adjacent directions from a given set S of directions that is symmetric with respect to the origin. Our results on Ordered Level Planarity imply NP -hardness for any S with ∣S∣ ≥ 4, even if the given graph is a matching. Manhattan Geodesic Planarity is the special case where S contains precisely the horizontal and vertical directions. Katz, Krug, Rutter, and Wolff claimed that Manhattan Geodesic Planarity can be solved in polynomial time for the special case of matchings [GD’09]. Our results imply that this is incorrect unless P = NP . Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Štefankovič. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity, and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli.

Heuristics for the Constrained Incremental Graph Drawing Problem

Visualization of information is a relevant topic in Computer Science, where graphs have become a standard representation model, and graph drawing is now a well-established area. Within this context, edge crossing minimization is a widely studied problem given its importance in obtaining readable representations of graphs. In this paper, we focus on the so-called incremental graph drawing problem, in which we try to preserve the user’s mental map when obtaining successive drawings of the same graph. In particular, we minimize the number of edge crossings while satisfying some constraints required to preserve the position of vertices with respect to previous drawings. We propose heuristic methods to obtain high-quality solutions to this optimization problem in the short computational times required for graph drawing applications. We also propose a mathematical programming formulation and obtain the optimal solution for small and medium instances. Our extensive experimentation shows the merit of our proposal with respect to both optimal solutions obtained with CPLEX and heuristic solutions obtained with LocalSolver, a well-known black-box solver in combinatorial optimization.

Temporal Treemaps: Static Visualization of Evolving Trees.

We consider temporally evolving trees with changing topology and data: tree nodes may persist for a time range, merge or split, and the associated data may change. Essentially, one can think of this as a time series of trees with a node correspondence per hierarchy level between consecutive time steps. Existing visualization approaches for such data include animated 2D treemaps, where the dynamically changing layout makes it difficult to observe the data in its entirety. We present a method to visualize this dynamic data in a static, nested, and space-filling visualization. This is based on two major contributions: First, the layout constitutes a graph drawing problem. We approach it for the entire time span at once using a combination of a heuristic and simulated annealing. Second, we propose a rendering that emphasizes the hierarchy through an adaption of the classic cushion treemaps. We showcase the wide range of applicability using data from feature tracking in time-dependent scalar fields, evolution of file system hierarchies, and world population.

Small Universal Point Sets for k-Outerplanar Graphs

A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ is universal for a class $$\mathcal G$$ of planar graphs if every graph of $$\mathcal{G}$$ has a planar straight-line embedding on $$\mathcal{S}$$ . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ -size universal point set in two distinct cases, namely when $$k=2$$ (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

Drawing planar graphs with many collinear vertices

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every $n$-vertex planar graph has a planar straight-line drawing with $\Omega(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^\sigma)$ collinear vertices, where $\sigma<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $\Theta(n)$ collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295 – 306]. Similar results are not possible for all bounded-treewidth planar graphs or for all bounded-degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

Root demotion: efficient post-processing of layered graphs to reduce dummy vertices for hierarchical graph drawing

One of the most popular hierarchical graph drawing frameworks - the Sugiyama, Tagawa, and Toda (STT) framework - often introduces dummy vertices to the given directed acyclic graph as part of its methodology to produce the final drawing. The inclusion of dummy vertices increases the size of the input graph and consequently inflates the run time of the subsequent algorithms. Given that most graph drawing problems are NP-hard, good layering algorithms or efficient post-processing - which produces a minimal number of dummy vertices - restrain the increased run time. Many layering or layering post-processing algorithms have quadratic or cubic run time. Here we introduce Root Demotion, a linear time post-processing algorithm for the layering produced by the Longest-Path algorithm that reduces the number of dummy vertices required compared to current post-processing algorithms, consequently shortening the remaining run time of the STT framework.

Variable neighborhood scatter search for the incremental graph drawing problem

Automated graph-drawing systems utilize procedures to place vertices and arcs in order to produce graphs with desired properties. Incremental or dynamic procedures are those that preserve key characteristics when updating an existing drawing. These methods are particularly useful in areas such as planning and logistics, where updates are frequent. We propose a procedure based on the scatter search methodology that is adapted to the incremental drawing problem in hierarchical graphs. These drawings can be used to represent any acyclic graph. Comprehensive computational experiments are used to test the efficiency and effectiveness of the proposed procedure.

Drawing graphs with mathematical programming and variable neighborhood search

In the Graph Drawing problem a set of symbols must be placed in a plane and their connections routed. To produce clear, easy to read diagrams, this problem is usually solved trying to minimize edges crossing and the area occupied, resulting in a NP-Hard problem. Our research focuses in drawing Entity Relationship (ER) diagrams, a challenging version of the problem where nodes have different sizes. Mathematical Programming models for the two solution phases, node placement and connection routing, are discussed and their exact resolution by an Integer Programming (IP) solver is evaluated. As the first phase proved to be specially hard to be solved exactly, a hybrid Variable Neighborhood Search (VNS) heuristic is proposed. Using IP techniques we obtained provably optimal (or close to optimal) solutions for the two different phases, at the expense of a large computational effort. We also show that our VNS based heuristic approach can produce close to optimal solutions in very short times for the hardest part of the solution process. Using either methods we have produced clearly better drawings than existing solutions.

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force-directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations. We formulate an abstract model of exact symbolic computation that augments algebraic computation trees with functions for computing radicals or roots of low-degree polynomials, and we show that this model cannot solve these graph drawing problems.

On multi-objective optimization aided drawing of special graphs

A special graph drawing problem related to the visualization of business process diagrams is considered. It is requested to find aesthetically looking paths between the given pairs of vertices of a grid, where the vertices represent the business process flow objects, and the paths represent the sequence flow. The sites of flow objects on the grid are fixed, and the sequence flow is defined. We state the problem of search for aesthetically looking paths as a problem of combinatorial multi-objective optimization, where the objectives correspond to the generally recognized criteria of aesthetics. To find a solution precisely, the algorithm of linear binary programming CPLEX was applied. For a faster solution, supposed for an interactive mode, a heuristic algorithm is developed. The experimental comparison of the mentioned algorithms is performed to substantiate the applicability of the latter.