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For some optimization problems on a graph \(G=(V,E)\), one can give a general formulation: Let \(c\colon E \to \mathbb{R}_{\geq 0}\) be a cost function on the edges and \(X \subseteq 2^E\) be a set of (so-called feasible) subsets of \(E\), one aims to minimize \(\sum_{e\in S} c(e)\) among all feasible \(S\in X\). This formulation covers, for instance, the shortest path problem by choosing \(X\) as the set of all paths between two vertices, or the minimum spanning tree problem by choosing \(X\) to be the set of all spanning trees. This bachelor thesis deals with a parametric version of this formulation, where the edge costs \(c_\lambda\colon E \to \mathbb{R}_{\geq 0}\) depend on a parameter \(\lambda\in\mathbb{R}_{\geq 0}\) in a concave and piecewise linear manner. The goal is to investigate the worst case minimum size of a so-called representation system \(R\subseteq X\), which contains for each scenario \(\lambda\in\mathbb{R}_{\geq 0}\) an optimal solution \(S(\lambda)\in R\). It turns out that only a pseudo-polynomial size can be ensured in general, but smaller systems have to exist in special cases. Moreover, methods are presented to find such small systems algorithmically. Finally, the notion of a representation system is relaxed in order to get smaller (i.e. polynomial) systems ensuring a certain approximation ratio.

This bachelor thesis is concerned with arrangements of hyperplanes, that
is, finite collections of hyperplanes in a finite-dimensional vector
space. Such arrangements can be studied using methods from
combinatorics, topology or algebraic geometry. Our focus lies on an
algebraic object associated to an arrangement \(\mathcal{A}\), the module \(\mathcal{D(A)}\) of
logarithmic derivations along \(\mathcal{A}\). It was introduced by K. Saito in the
context of singularity theory, and intensively studied by Terao and
others. If \(\mathcal{D(A)}\) admits a basis, the arrangement \(\mathcal{A}\) is called free.
Ziegler generalized the concept of freeness to so-called
multiarrangements, where each hyperplane carries a multiplicity. Terao
conjectured that freeness of arrangements can be decided based on the
combinatorics. We pursue the analogous question for multiarrangements in
special cases. Firstly, we give a new proof of a result of Ziegler
stating that generic multiarrangements are totally non-free, that is,
non-free for any multiplicity. Our proof relies on the new concept of
unbalanced multiplicities. Secondly, we consider freeness asymptotically
for increasing multiplicity of a fixed hyperplane. We give an explicit
bound for the multiplicity where the freeness property has stabilized.