arXiv subjects

Michael Kaufmann

Publications and source records attributed to Michael Kaufmann.

At least 37 records · Page 2Linked to original sources

arXiv2023

An Online Framework to Interact and Efficiently Compute Linear Layouts of Graphs

We present a prototype online system to automate the procedure of computing different types of linear layouts of graphs under different user-specific constraints. Currently, four different types of linear layouts are supported: stack, queue, rique and deque, as well as, any mixture of them. The system consists of two main components; the client and the server sides. The client side is built upon an easy-to-use editor, which supports basic interaction with graphs, enriched with several additional features to allow the user to define and further constraint the linear layout to be computed. The server side, which is available to multiple clients through a well-documented API, is responsible for the actual computation of the linear layout. Its algorithmic core is an extension of a SAT formulation that is known to be robust enough to solve non-trivial instances in reasonable amount of time.

cs.DM↗

arXiv2023

$k$-planar Placement and Packing of $Δ$-regular Caterpillars

This paper studies a \emph{packing} problem in the so-called beyond-planar setting, that is when the host graph is ``almost-planar'' in some sense. Precisely, we consider the case that the host graph is $k$-planar, i.e., it admits an embedding with at most $k$ crossings per edge, and focus on families of $Δ$-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree $Δ$. We study the dependency of $k$ from the number $h$ of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called \emph{placement}) and when they are not. We give necessary and sufficient conditions for the placement of $h$ $Δ$-regular caterpillars and sufficient conditions for the packing of a set of $Δ_1$-, $Δ_2$-, $\dots$, $Δ_h$-regular caterpillars such that the degree $Δ_i$ and the degree $Δ_j$ of the non-leaf vertices can differ from one caterpillar to another, for $1 \leq i,j \leq h$, $i\neq j$.

math.CO↗

arXiv2022

RAC Drawings of Graphs with Low Degree

Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90° angles. In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs.

cs.CG↗

arXiv2022

Computing Bend-Minimum Orthogonal Drawings of Plane Series-Parallel Graphs in Linear Time

A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal and vertical segments between its end-points. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term "plane" indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper, we positively answer the question above for the widely-studied class of series-parallel graphs. Our linear-time algorithm is based on a characterization of the planar series-parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is "rolled-up" in an orthogonal drawing of the graph.

cs.CG↗

arXiv2022

Graph Product Structure for h-Framed Graphs

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020]. In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size $3\lfloor h/2 \rfloor +\lfloor h/3 \rfloor -1$. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 495 to 81 and from 32225k(k-3) to 61k, respectively. We also employ the product structure machinery to improve the current upper bounds of twin-width of planar and 1-planar graphs from 183 to 37, and from O(1) to 80, respectively. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.

cs.DS↗

arXiv2022

The Lokahi Prototype: Toward the automatic Extraction of Entity Relationship Models from Text

Entity relationship extraction envisions the automatic generation of semantic data models from collections of text, by automatic recognition of entities, by association of entities to form relationships, and by classifying these instances to assign them to entity sets (or classes) and relationship sets (or associations). As a first step in this direction, the Lokahi prototype can extract entities based on the TF*IDF measure, and generate semantic relationships based on document-level co-occurrence statistics, for example with likelihood ratios and pointwise mutual information. This paper presents results of an explorative, prototypical, qualitative and synthetic research, summarizes insights from two research projects and, based on this, indicates an outline for further research in the field of entity relationship extraction from text.

cs.IR↗

arXiv2021

The number of crossings in multigraphs with no empty lens

Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of computing series, 1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in $G$ is at least $αe^3/n^2$, for a suitable constant $α>0$. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in $G$ is at least $αe^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.

math.CO↗

arXiv2021

Spirality and Rectilinear Planarity Testing of Independent-Parallel SP-Graphs

We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of ``rectilinear shapes'' for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.

cs.DS↗

arXiv2021

Recognizing and Embedding Simple Optimal 2-Planar Graphs

In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs, the recognition problem has been settled, namely it is NP-complete for the general case, while optimal 1-planar graphs, i.e. those with maximum density, can be recognized in linear time. For 2-planar graphs, the picture is less complete. As expected, the recognition problem has been found to be NP-complete in general. In this paper, we consider the recognition of simple optimal 2-planar graphs. We exploit a combinatorial characterization of such graphs and present a linear time algorithm for recognition and embedding.

cs.DS↗

arXiv2021

The Density of Fan-Planar Graphs

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common. Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Turán-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics. As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of work on nearly-planar graphs from the area of graph drawing and combinatorial embeddings. In this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form the largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly.

cs.DM↗

arXiv2021

The Mixed Page Number of Graphs

A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in a linear layout. Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. In this work we initiate the study of the mixed page number of a graph which corresponds to the minimum number of such sets. First, we study the edge density of graphs with bounded mixed page number. Then, we focus on complete and complete bipartite graphs, for which we derive lower and upper bounds on their mixed page number. Our findings indicate that combining stacks and queues is more powerful in various ways compared to the two traditional layouts.

cs.DS↗

arXiv2021

Efficient and Accurate In-Database Machine Learning with SQL Code Generation in Python

Following an analysis of the advantages of SQL-based Machine Learning (ML) and a short literature survey of the field, we describe a novel method for In-Database Machine Learning (IDBML). We contribute a process for SQL-code generation in Python using template macros in Jinja2 as well as the prototype implementation of the process. We describe our implementation of the process to compute multidimensional histogram (MDH) probability estimation in SQL. For this, we contribute and implement a novel discretization method called equal quantized rank binning (EQRB) and equal-width binning (EWB). Based on this, we provide data gathered in a benchmarking experiment for the quantitative empirical evaluation of our method and system using the Covertype dataset. We measured accuracy and computation time and compared it to Scikit Learn state of the art classification algorithms. Using EWB, our multidimensional probability estimation was the fastest of all tested algorithms, while being only 1-2% less accurate than the best state of the art methods found (decision trees and random forests). Our method was significantly more accurate than Naive Bayes, which assumes independent one-dimensional probabilities and/or densities. Also, our method was significantly more accurate and faster than logistic regression. This motivates for further research in accuracy improvement and in IDBML with SQL code generation for big data and larger-than-memory datasets.

cs.DB↗

arXiv2021

Rectilinear Planarity Testing of Plane Series-Parallel Graphs in Linear Time

A plane graph is rectilinear planar if it admits an embedding-preserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal $O(n)$ time for any plane series-parallel graph $G$ with $n$ vertices. If $G$ is rectilinear planar, an embedding-preserving rectilinear planar drawing of $G$ can be constructed in $O(n)$ time. Our result is based on a characterization of rectilinear planar series-parallel graphs in terms of intervals of orthogonal spirality that their components can have, and it leads to an algorithm that can be easily implemented.

cs.DS↗

arXiv2020

Lazy Queue Layouts of Posets

We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w. The conjecture has been confirmed for posets of width w=2 via so-called lazy linear extension. We extend and thoroughly analyze lazy linear extensions for posets of width w > 2. Our analysis implies an upper bound of $(w-1)^2 +1$ on the queue number of width-w posets, which is tight for the strategy and yields an improvement over the previously best-known bound. Further, we provide an example of a poset that requires at least w+1 queues in every linear extension, thereby disproving the conjecture for posets of width w > 2.

cs.DS↗

arXiv2020

Four Pages Are Indeed Necessary for Planar Graphs

An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by demonstrating planar graphs that require four pages in any of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.

cs.DS↗

arXiv2020

Monotone Arc Diagrams with few Biarcs

We show that every planar graph can be represented by a monotone topological 2-page book embedding where at most 15n/16 (of potentially 3n-6) edges cross the spine exactly once.

cs.CG↗

arXiv2020

On Layered Fan-Planar Graph Drawings

In this paper, we study fan-planar drawings that use $h$ layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter tractable in $h$, via a reduction to a similar result for planar graphs by Dujmović et al. If the embedding is not fixed, then we give partial results for $h=2$: It was already known how to test existence of fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how to test this for trees. Along the way, we exhibit other interesting results for graphs with a fan-planar proper $h$-layer drawings; in particular we bound their pathwidth and show that they have a bar-1-visibility representation.

cs.CG↗

arXiv2019

Coloring outerplanar graphs and planar 3-trees with small monochromatic components

In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.

cs.DS↗