Abstract
Good treewidth lower bounds can be used in branch-and-bound methods.
The better and faster the bounds, the faster the branch-and-bound algorithm.
They are also useful to estimate the running time of
a dynamic programming algorithm based on tree-decompositions.
A large treewidth lower bound indicates that the treewidth is large.
Because of the exponential
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influence of the treewidth on the running time of such methods,
there is only little hope to find an efficient algorithm based on tree-decompositions.
Treewidth lower bounds in connection with treewidth upper bounds can help to assess
the quality of these bounds.
A small gap between the bounds means tight bounds, and a large gap indicates room for improvements.
Communication networks are an important part of our world.
Real networks consist of elements that are not infallible.
They can be modelled by graphs,
where we associate to each vertex a rational number
that is the reliability of the corresponding network element.
We assume that the reliabilities of the elements are stochastically independent.
A very crucial issue for designing and maintaining networks is their reliability.
The notion network reliability addresses questions such as:
'What is the probability that two distinguished sites can communicate,
while parts of the network broke down?'
Computing the network reliability is in general NP-hard.
In this thesis, we develop new treewidth lower bounds.
All new lower bounds have in common that they are based on a combination of
existing (degree-based) lower bounds and edge contractions.
Contracting an edge is the operation that replaces an edge and its two endpoints
by one vertex that is made adjacent to all the neighbours of the two endpoint of the contracted edge.
A minor of a graph is a graph obtained from a subgraph by contracting edges.
The main idea to improve treewidth lower bounds
is to take an existing lower bound over all subgraphs or minors.
In this way, we obtain a number of parameters, all treewidth lower bounds,
study relations between them and their computational complexity.
For the parameters that are NP-hard to compute we develop heuristics.
In experiments, we compare the quality of the lower bounds and their running times amongst each other
and also to the best known treewidth upper bounds for a number of graphs
from various areas such as probabilistic networks or frequency assignment problems.
The results of the experiments made very clear
that combining edge contraction with existing treewidth lower bounds
is a very vital idea to improve upon treewidth lower bounds.
One treewidth lower bound parameter is the contraction degeneracy.
It is the maximum over all minors of a graph of the minimum degree of the minor.
This parameter is NP-hard to compute.
Due to its elementary character, it is an interesting study object in its own right
and not only as a treewidth lower bound.
We present a polynomial time method for computing the contraction degeneracy
based on dynamic programming for cographs.
We also present a framework for network reliability problems for graphs of bounded treewidth.
In our model we have two distinguished sets of vertices (severs and clients).
With the framework we can answer more questions, such as
'What is the probability that every client is connected to at least one server?' or
'What is the expected number of connected components with at least one server?'.
These and similar question are proven to be #P-hard.
However, using our framework they can be solved in polynomial time on graphs of bounded treewidth.
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