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Given points in the plane, connect them using minimum ink. Though the task seems simple, it turns out to be very time consuming. In fact, scientists believe that computers cannot efficiently solve it. So, do we have to resign? This book examines such NP-hard network-design problems, from connectivity problems in graphs to polygonal drawing problems on the plane. First, we observe why it is so hard to optimally solve these problems. Then, we go over to attack them anyway. We develop fast algorithms that find approximate solutions that are very close to the optimal ones. Hence, connecting points with slightly more ink is not hard.
The complexity of membership problems for finite recurrent systems and minimal triangulations
(2006)
The dissertation thesis studies the complexity of membership problems. Generally, membership problems consider the question whether a given object belongs to a set. Object and set are part of the input. The thesis studies the complexity of membership problems for two special kinds of sets. The first problem class asks whether a given natural number belongs to a set of natural numbers. The set of natural numbers is defined via finite recurrent systems: sets are built by iterative application of operations, like union, intersection, complementation and arithmetical operations, to already defined sets. This general problem implies further problems by restricting the set of used operations. The thesis contains completeness results for well-known complexity classes as well as undecidability results for these problems. The second problem class asks whether a given graph is a minimal triangulation of another graph. A graph is a triangulation of another graph, if it is a chordal spanning supergraph of the second graph. If no proper supergraph of the first graph is a triangulation of the second graph, the first graph is a minimal triangulation of the second graph. The complexity of the membership problem for minimal triangulations of several graph classes is investigated. Restricted variants are solved by linear-time algorithms. These algorithms rely on appropriate characterisations of minimal triangulations.
Graphs provide a key means to model relationships between entities.
They consist of vertices representing the entities,
and edges representing relationships between pairs of entities.
To make people conceive the structure of a graph,
it is almost inevitable to visualize the graph.
We call such a visualization a graph drawing.
Moreover, we have a straight-line graph drawing
if each vertex is represented as a point
(or a small geometric object, e.g., a rectangle)
and each edge is represented as a line segment between its two vertices.
A polyline is a very simple straight-line graph drawing,
where the vertices form a sequence according to which the vertices are connected by edges.
An example of a polyline in practice is a GPS trajectory.
The underlying road network, in turn, can be modeled as a graph.
This book addresses problems that arise
when working with straight-line graph drawings and polylines.
In particular, we study algorithms
for recognizing certain graphs representable with line segments,
for generating straight-line graph drawings,
and for abstracting polylines.
In the first part, we first examine,
how and in which time we can decide
whether a given graph is a stick graph,
that is, whether its vertices can be represented as
vertical and horizontal line segments on a diagonal line,
which intersect if and only if there is an edge between them.
We then consider the visual complexity of graphs.
Specifically, we investigate, for certain classes of graphs,
how many line segments are necessary for any straight-line graph drawing,
and whether three (or more) different slopes of the line segments
are sufficient to draw all edges.
Last, we study the question,
how to assign (ordered) colors to the vertices of a graph
with both directed and undirected edges
such that no neighboring vertices get the same color
and colors are ascending along directed edges.
Here, the special property of the considered graph is
that the vertices can be represented as intervals
that overlap if and only if there is an edge between them.
The latter problem is motivated by an application
in automated drawing of cable plans with vertical and horizontal line segments,
which we cover in the second part.
We describe an algorithm that
gets the abstract description of a cable plan as input,
and generates a drawing that takes into account
the special properties of these cable plans,
like plugs and groups of wires.
We then experimentally evaluate the quality of the resulting drawings.
In the third part, we study the problem of abstracting (or simplifying)
a single polyline and a bundle of polylines.
In this problem, the objective is to remove as many vertices as possible from the given polyline(s)
while keeping each resulting polyline sufficiently similar to its original course
(according to a given similarity measure).