LNCS

Let C={C1,...,Cn} denote a collection of translates of a regular convex k-gon in the plane with the stacking order. The collection C forms a visibility clique if for everyi < j the intersection Ci and (Ci ∩ Cj)\⋃i<l<jCl =∅.elements that are stacked between them, i.e., We show that if C forms a visibility clique its size is bounded from above by O(k4) thereby improving the upper bound of 22k from the aforementioned paper. We also obtain an upper bound of 22(k/2)+2 on the size of a visibility clique for homothetes of a convex (not necessarily regular) k-gon

Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

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 is realized as a polygonal path composed of line segments with two adjacent directions from a given set $S$ of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply $NP$-hardness for any $S$ with $|S|\ge 4$ even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where $S$ contains precisely the horizontal and vertical directions, can be solved in polynomial time [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 \v{S}tefankovi\v{c}. 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.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Embedding Stacked Polytopes on a Polynomial-Size Grid

We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) x O(n4) x O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by G unter M. Ziegler, and is the rst nontrivial subexponential upper bound on the long-standing open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about effcient versions of Steinitz's 1916 theorem. An immediate consequence of our result is that O(log n)-bit coordinates suffice for a greedy routing strategy in planar 3-trees.Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 2458/1-1

Convex Hulls of Random Order Types

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average 4-8/(n^2 - n +2) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d+o(1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods allow to show the following relative of the Erd?s-Szekeres theorem: for any fixed k, as n ? ?, a proportion 1 - O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the 2-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral or one of A?, S? or A? (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein

Descriptive complexity of real computation and probabilistic independence logic

We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in non-deterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable union of closed sets in the usual topology of R^n. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class existsR (a class of problems polynomial time reducible to the true existential theory of the reals) and hence lies between NP and PSPACE. Finally we apply our results to determine the data complexity of probabilistic independence logic.Peer reviewe

Tractability Frontiers in Probabilistic Team Semantics and Existential Second-Order Logic over the Reals

Peer reviewe

Influence of diagram layout and scrolling on understandability of BPMN processes: an eye tracking experiment with BPMN diagrams

Business process modeling is an important activity for developing software systems—especially within digitization projects and when realizing digital business models. Specifying requirements and building executable workflows is often done by using BPMN 2.0 process models. Although there are several style guides available for BPMN, e.g., by Silver and Richard (BPMN method and style, vol 2, Cody-Cassidy Press, Aptos, 2009), there has not been much empirical research done into the consequences of the diagram layout. In particular, layouts that require scrolling have not been investigated yet. The aim of this research is to establish layout guidelines for business process modeling that help business process modelers to create more understandable business process diagrams. For establishing benefits and penalties of different layouts, a controlled eye tracking experiment was conducted, in which data of 21 professional software developers was used. Our results show that horizontal layouts are less demanding and that as many diagram elements as possible should be put on the initially visible screen area because such diagram elements are viewed more often and longer. Additionally, diagram elements related to the reader’s task are read more often than those not relevant to the task. BPMN modelers should favor a horizontal layout and use a more complex snake or multi-line layout whenever the diagrams are too large to fit on one page in order to support BPMN model comprehension

Facets of Planar Graph Drawing

This thesis makes a contribution to the field of Graph Drawing, with a focus on the planarity drawing convention. The following three problems are considered. (1) Ordered 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. With reductions from Ordered Level Planarity, we show NP-hardness of multiple problems whose complexity was previously open, and strengthen several previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two nontrivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli [2015]. We settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Stefankovic [2013]. We also present a reduction to Manhattan Geodesic Planarity, showing that a previously [2009] claimed polynomial time algorithm is incorrect unless P=NP. (2) Two-page book embeddings of triconnected planar graphs: We show that every triconnected planar graph of maximum degree five is a subgraph of a Hamiltonian planar graph or, equivalently, it admits a two-page book embedding. In fact, our result is more general: we only require vertices of separating 3-cycles to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: we describe a family of triconnected planar graphs that cannot be realized on two pages and where every vertex of a separating 3-cycle has degree at most six. Our results strengthen earlier work by Heath [1995] and by Bauernöppel [1987] and, independently, Bekos, Gronemann, and Raftopoulou [2016], who showed that planar graphs of maximum degree three and four, respectively, can always be realized on two pages. The proof is constructive and yields a quadratic time algorithm to realize the given graph on two pages. (3) Convexity-increasing morphs: We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal.Diese Arbeit behandelt drei unterschiedliche Problemstellungen aus der Disziplin des Graphenzeichnens (Graph Drawing). Bei jedem der behandelten Probleme ist die gesuchte Darstellung planar. (1) Ordered Level Planarity: Wir führen das Problem Ordered Level Planarity ein, bei dem es darum geht, einen Graph so zu zeichnen, dass jeder Knoten an einer vorgegebenen Position der Ebene platziert wird und die Kanten als y-monotone Kurven dargestellt werden. Dies kann als eine Variante von Level Planarity interpretiert werden, bei der die Knoten jedes Levels in einer vorgeschriebenen Reihenfolge platziert werden müssen. Wir klassifizieren die Eingaben bezüglich ihrer Komplexität in Abhängigkeit von sowohl dem Maximalgrad, als auch der maximalen Anzahl von Knoten, die demselben Level zugeordnet sind. Wir motivieren die Ergebnisse, indem wir Verbindungen zu einigen anderen Graph Drawing Problemen herleiten: Mittels Reduktionen von Ordered Level Planarity zeigen wir die NP-Schwere einiger Probleme, deren Komplexität bislang offen war. Insbesondere wird gezeigt, dass Clustered Level Planarity bereits für Instanzen mit zwei nichttrivialen Clustern NP-schwer ist, was eine Frage von Angelini, Da Lozzo, Di Battista, Frati und Roselli [2015] beantwortet. Wir zeigen die NP-Schwere des Bi-Monotonicity Problems und beantworten damit eine Frage von Fulek, Pelsmajer, Schaefer und Stefankovic [2013]. Außerdem wird eine Reduktion zu Manhattan Geodesic Planarity angegeben. Dies zeigt, dass ein bestehender [2009] Polynomialzeitalgorithmus für dieses Problem inkorrekt ist, es sei denn, dass P=NP ist. (2) Bucheinbettungen von dreifach zusammenhängenden planaren Graphen mit zwei Seiten: Wir zeigen, dass jeder dreifach zusammenhängende planare Graph mit Maximalgrad 5 Teilgraph eines Hamiltonischen planaren Graphen ist. Dies ist äquivalent dazu, dass ein solcher Graph eine Bucheinbettung auf zwei Seiten hat. Der Beweis ist konstruktiv und zeigt in der Tat sogar, dass es für die Realisierbarkeit nur notwendig ist, den Grad von Knoten separierender 3-Kreise zu beschränken - die übrigen Knoten können beliebig hohe Grade aufweisen. Dieses Ergebnis ist bestmöglich: Wenn die Gradschranke auf 6 abgeschwächt wird, gibt es Gegenbeispiele. Diese Ergebnisse verbessern Resultate von Heath [1995] und von Bauernöppel [1987] und, unabhängig davon, Bekos, Gronemann und Raftopoulou [2016], die gezeigt haben, dass planare Graphen mit Maximalgrad 3 beziehungsweise 4 auf zwei Seiten realisiert werden können. (3) Konvexitätssteigernde Deformationen: Wir zeigen, dass jede planare geradlinige Zeichnung eines intern dreifach zusammenhängenden planaren Graphen stetig zu einer solchen deformiert werden kann, in der jede Fläche ein konvexes Polygon ist. Dabei erhält die Deformation die Planarität und ist konvexitätssteigernd - sobald ein Winkel konvex ist, bleibt er konvex. Wir geben einen effizienten Algorithmus an, der eine solche Deformation berechnet, die aus einer asymptotisch optimalen Anzahl von Schritten besteht. In jedem Schritt bewegen sich entweder alle Knoten entlang horizontaler oder entlang vertikaler Geraden