Networks are present in our lives in numerous different environments: to name just a few, networks can model social
relationships, they can model the Internet and links between web pages, they might model the spread of a virus infection
between people, and they might represent computer processors/sensors that have to exchange information.
This project aims to obtain new insights into the behaviour of networks, which are studied from a geometric and
computational perspective. Thereto, the project brings together researchers from different areas such as computational
geometry, discrete mathematics, graph drawing, and probability. Among of the topics of research are enumerative problems
on geometric networks, crossing numbers, random networks, imprecise models of data, restricted orientation geometry.
Combinatorial approaches are combined with algorithms. Algorithmic applications of networks are also studied in the context
of unmanned aerial vehicles (UAVs) and in the context of musical information retrieval (MIR).
The project contains the work packages: “Geometric networks”, "Stochastic Geometry and Networks", “Restricted orientation
geometry”, “Graph-based algorithms for UAVs and for MIR”, and “Dissemination and gender equality promotion”.
The project connects researchers from 14 universities located in Austria, Belgium, Canada, Chile, Czech Republic, Italy,
Mexico, and Spain, who will collaborate and share their different expertise in order to obtain new knowledge on the
combinatorics of networks and applications.
ComPoSe — Combinatorics of Point Sets and Arrangements of Objects
This CRP focuses on combinatorial properties of discrete sets of
points and other simple geometric objects primarily in the plane. In
general, geometric graphs are a central topic in discrete and
computational geometry, and many important questions in mathematics
and computer science can be formulated as problems on geometric
graphs. In the current context, several families of geometric graphs,
such as proximity and skeletal structures, constitute useful
abstractions for the study of combinatorial properties of the point
sets on which they are defined. For arrangements of other objects,
such as lines or convex sets, their combinatorial properties are
usually also described via an underlying graph structure.
The following four tasks are well-known hard problems in this area and
will form the backbone of the current project. We will consider the
intriguing class of Erdős-Szekeres type problems, variants of
graph problems with colored vertices, counting and enumeration
problems for specific classes of geometric graphs, and generalizations
of order types as a versatile tool to investigate the combinatorics of
point sets. All these problems are combinatorial problems on geometric
graphs and are interrelated in the sense that approaches developed for
one of them will also be useful for the others. Moreover, progress in
one direction might provide a better understanding for related
questions. Our main objective is to gain deeper insight into the
structure of this type of problems and to contribute major steps
towards their final solution.
Erdős-Szekeres problems. We will investigate specific
variants of this famous group of problems, such as colored versions,
and use newly developed techniques, such as a recent generalized
notion of convexity, to progress on this topic. A typical example is
the convex monochromatic quadrilateral problem in Section (iv) of the
Call for Outline Proposals: Prove or disprove that every (sufficiently
large) bichromatic point set contains an empty convex monochromatic
quadrilateral. Recent progress on this and other Erdős-Szekeres
type problems has been made by the PIs Aichholzer, Hurtado, Pach,
Valtr, and Welzl.
Colored point sets. An interesting family of questions is the
existence of constrained colorings of point sets. We may consider, for
instance, the problem of coloring a set of points in a way such that
any unit disk with sufficiently many points contains all colors. Also,
colored versions of classical Helly-type results continue to be a
source of fundamental problems, requiring the use of combinatorial and
topological tools. In particular we are interested in colored
versions of Tverberg-type results and their generalization of
Tverberg-Vrećica-type. Pach founded the class of ‘covering
colored sets’ problems and will cooperate on these problems with
Cardinal and Felsner in particular, but also with all other PIs.
Counting, enumerating, and sampling of crossing-free
configurations. Planar graphs are a core topic in abstract graph
theory. Their counterpart in geometric graph theory are crossing-free
(plane) graphs. Interesting questions arise from considering specific
classes of plane graphs, such as triangulations, spanning cycles,
spanning trees, and matchings. For example, the flip-graph of the set
of all graphs of a given class allows a fast enumeration of all
elements from this class and even efficient optimization with respect
to certain criteria. But when it comes to more intricate demands, like
counting or sampling a random element, very little is understood. We
will put emphasis on counting, enumerating, and sampling methods for
several of the mentioned graph classes. Related extremal results
(e.g. upper bounds on the number of triangulations) will also be
considered for other classes, like string graphs of a fixed order k
(intersection graphs of curves in the plane with at most k
intersections per pair) or visibility graphs in the presence of at
most k connected obstacles. Aichholzer, Hurtado, and Welzl have been
involved in recent progress on lower and upper bounds for the number
of several mentioned classes of geometric graphs and will cooperate
with Pach (intersection graphs), Valtr, and Felsner (higher
dimensions) on enumerating and counting.
Order types (rank 3 oriented matroids). Order types play a
central role in the above mentioned problems, and constitute a useful
tool to investigate the combinatorics of point sets. This is done,
e.g., by providing small instances of vertex sets for extremal
geometric graphs in enumeration problems. Our goal is to generalize,
and at the same time specialize, this concept. For example, we plan to
investigate the k-set problem as well as a generalization of the
Erdős-Szekeres theorem for families of convex bodies in the
plane. Typically, progress on the k-set problem has frequently been
achieved in the language of pseudoline arrangements, which are dual to
order types. In particular we are interested in combinatorial results
ranging from Sylvester-type results to counting certain cells, and the
number and structure of arrangements of n pseudo-lines. Felsner is an
expert on pseudo-line arrangements and will collaborate here with
Valtr, Pach, Welzl and Aichholzer on order types. Moreover all PIs
have been working on the kset problem individually and will make a
joint afford.
This CRP tackles fundamental questions at the intersection of
mathematics and theoretical computer science. It is well known that in
this area some problems require only days to be solved, others may
take decades or even more. Thus, the working schedule with respect to
obtaining the desired theoretical results must follow the standard
approach: Continuation of work in progress, evaluation of the results
obtained by other authors and groups, and continuous identification of
new directions for progress and exploration, hence always advancing
the frontiers of knowledge. Since it is infeasible to impose a proper
temporal order on the objectives and milestones to be attained - the
conceptual implications are manifold, and many of the stated
objectives are strongly interrelated - it will be the very progress of
research and the obtained results that mark our progress in time. This
is guaranteed by the competence of the team. The major 'visible'
milestones will be the regular presentations of joint papers in the
main conferences of the field, the corresponding submissions to
journals, and a series of progress reports that will help in keeping a
clear and consistent guidance and interaction with the other teams.
Several of the mentioned problems are long-standing open questions and
known to be hard. Therefore we will consider several specific variants
of them to determine how far state-of-the-art methods can be used and
where new approaches have to be found. This will definitely improve
our understanding of the structure of these problems, with the goal of
making major contributions towards their solution or, in the ideal
case, to finally settle them. Most of our approaches will be of
theoretical nature. But we will also make intensive use of computers
for enumeration and experiments, to get initial insights into the
structure of problems, or to support or refute conjectures.
It is well known that the mentioned problems have resisted several
previous attacks and therefore require the cooperation of researchers
with strong and complementary expertise. We consider large-scale
collaboration on these topics as one of the main ingredients for
success. Thus we will not have individual projects running in
parallel, but all participants will jointly work on the topics, in a
massive collaborative effort. To guarantee a strong interaction
between the members of the group we will maintain regular exchanges of
senior researchers and students, regular joint research workshops (1-2
per year), and frequent visits.
Industrial Geometry - An emerging research area
Industrial geometry is based on computational techniques which
originated in various areas of applied geometry. To give a few
examples, the methods of Computer Aided Geometric Design form the
mathematical foundation of the powerful CAD technologies available
today. Computer Vision provides methods for inspecting and analyzing
images and videos. Image processing is used to reconstruct geometric
features from digital image data, such as X-ray or computer tomography
images. Computational Geometry provides efficient algorithms for
solving fundamental geometrical problems. Robot kinematics deals with
geometric problems which occur in connection with robot mechanical
systems.
In recent years, these different areas of research have started to
become increasingly interconnected, and begun even to merge. A driving
force in this process is the increasing complexity of applications,
where one field of research alone would be insufficient to achieve
useful results. Novel technologies for acquisition and processing of
data lead to new and increasingly challenging problems, whose solution
requires the combination of techniques from different branches of
applied geometry.
Besides Subproject S09201: Coordination and Service, Knowledge transfer
and sustainability, TU Graz hosts the following subprojects
Subproject S09205 (Computational Geometry)
(Principal Investigator: Oswin Aichholzer)
Computational Geometry is dedicated to the algorithmic study of
elementary geometric questions. Traditionally it deals with basic
geometric objects like points, lines, and planes. For real world
applications, however, often reliable techniques for advanced
geometric primitives like surfaces and location query structures are
needed.
The role of this project is twofold. On one hand it will provide the
theoretical background for advanced geometric algorithms and data
structures for several other projects within this joint research
project (JRP). These include geometric structures for fast information
retrieval, the generation and manipulation of triangular meshes, the
computation of suitable distance functions to multidimensional
objects, and the representation of advanced geometric objects.
Another aim of this project is to develop novel techniques for the
manipulation and optimization of geometric structures. Here the
emphasis is on geometric graphs (triangulation-like and Voronoi
diagram-like structures, spanning trees). Properties of these
structures will be investigated, with the goal of designing more
efficient geometric algorithms and data structures.
Existing geometric algorithms libraries (CGAL, LEDA) will be used to
guarantee robustness of the developed algorithms.
Subproject S09209 (Computational Differential Geometry)
(Principal investigator: Johannes Wallner)
Computational Differential Geometry means methods of both numerical and
discrete mathematics with the purpose of investigating and modeling curves
and surfaces. The main theme of this research project is the robust
analysis of differential properties of surfaces, the creation of discrete
and semi-discrete models of freeform surfaces, and the study of geometric
properties of such models. It is only recently that the wealth of
interesting geometry connected to applications in, say, architecture, has
come to the attention of mathematicians, and presumably only a small part of it
has been investigated. We are investigating topics of Discrete Differential
Geometry: discrete curvatures based on parallel meshes, quad-based and hex-based
discrete surfaces, Christoffel duality, and others. New lines of research of
semi-discrete surfaces and inverse problems in connection with integral
invariants.
Geometric objects such as points, lines and polygons are the key elements of a big variety of interesting research problems in computer science. With the rise of modern technologies, more and more of these tasks are solved by computers, as opposed to the classic pen-and-paper approach.
Over the last thirty years, researchers around the world have developed different techniques and algorithms that take advantage of the structure provided by geometry to sIn this thesis we consider triangles in the colored Euclidean plane.
We call a triangle monochromatic if all its vertices have the same color.
First, we study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a monochromatic translated copy of the triangle.
Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is similar to the given triangle.
Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring.
We also show, that for every triangle there exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic.
Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points with distance 1, which have the same color.
olve these problems.
This area of research, in between mathematics and computer science, is known as discrete and computational geometry. In this joint seminar we plan to use tools from discrete aIn this thesis we consider triangles in the colored Euclidean plane.
We call a triangle monochromatic if all its vertices have the same color.
First, we study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a monochromatic translated copy of the triangle.
Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is similar to the given triangle.
Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring.
We also show, that for every triangle there exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic.
Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points with distance 1, which have the same color.
Computational geometry (such as order-type-like properties, see below) and apply them to problems that
come motivated from the field of sensor networks.
Zentrales Thema dieser gemeinsamen Forschung ist die Untersuchung
grundlegender Datenstrukturen aus dem Bereich der rechnerischen
Geometrie (Computational Geometry), einem relativ jungen Teilgebiet
der (theoretischen) Informatik. Dabei sollen sowohl theoretische
Aspekte untersucht werden, als auch deren konkrete Umsetzung im Rahmen
einer allgemein verwendbaren Programmbibliothek.
Auf Seite der theoretischen Untersuchungen sollen sowohl klassische
Datenstrukturen, wie Voronoi-Diagramme oder Triangulierungen, aber auch
relativ neue Datenstrukturen, wie Pseudo-Triangulierungen oder
Straight-Skeletons, untersucht werden. Genauere Einzelheiten werden in
den nachfolgenden Abschnitten beschrieben. Gleichzeitig soll aber auch
für alle untersuchten Strukturen deren tatsächliche
Verwendbarkeit in der Praxis berücksichtigt werden. So ist es
geplant, jeweils spezielle Aspekte dieser Strukturen in konkreten
Implementationen umzusetzen. Um einen bestmöglichen Nutzen der
erzielten Ergebnisse zu gewährleisten, sollen die Implementationen
in einer standardisierten Bibliothek der gesamten CG-Community zur
Verfügung gestellt werden. Dazu ist die Umsetzung in CGAL
(Computational Geometry Algorithms Library, siehe { www.cgal.org})
geplant. Es handelt sich dabei um ein ursprünglich von
der EU gefördertes Projekt, das insbesondere von unserem
französichen Projektpartner an zentraler Stelle mitbetrieben wird.
The general topic of this project is the investigation of geometric
graphs, i.e., graphs where the vertex set is a point set in the
plane and the edges are straight line segments spanned by these points.
Throughout we assume the points to be in general position, that is no
three of them lie on a common line, and to be labeled.
Geometric graphs are a versatile data structure, as they include
triangulations, Euclidean spanning trees, spanning paths,
polygonalizations, plane perfect matchings and so on. The
investigation of geometric graphs belongs to the field of
(combinatorial) mathematics, graph theory, as well as to discrete and
computational geometry. The alliance of our two research groups will
perfectly cover these fields. For example this will allow us to use
an interesting combination of enumerative investigations (lead by the
Austrian team) and theoretical research (coordinated by the Spanish
group). Let us point out that this combination of theoretical
knowledge and practical experience, which is perfectly provided by the
combination of these two teams, will be essential for the success of
this project.
There are many classic as well as new tantalizing problems on
geometric graphs, and the investigation of seemingly unrelated
questions often leads to new relations and deeper structural insight.
So the focus of this project is to investigate several classes of
problems with the common goal of optimizing properties of geometric graphs.