Convex Optimization Techniques for Geometric Covering Problems (in German)
Jan Hendrik Rolfes
Synopsis "Convex Optimization Techniques for Geometric Covering Problems (in German)"
The present thesis is a commencement of a generalization of covering results in specific settings, such as the Euclidean space or the sphere, to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(X, d)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$, denoted by $\Ncal(X, r)$. For finite $X$ this problem coincides with an instance of the combinatorial \textsc{set cover} problem, which is $\mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for $\Ncal(X, r)$. \\ The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv\'atal's algorithm for the weighted case of \textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S n$ and retrieve the best non-asymptotic bound of B\"or\"oczky and Wintsche. Additionally, the algorithm can be used to determine coverings of Euclidean space with arbitrary measurable objects having non-empty interior. The quality of these coverings slightly improves a bound of Nasz\'odi. \\ For the lower bounds we develop a sequence of bounds $\Ncal t(X, r)$ that converge after finitely (say $\alpha\in\N$) many steps: $$\Ncal 1(X, r)\leq \ldots \leq \Ncal \alpha(X, r)=\Ncal(X, r).$$ The drawback of this sequence is that the bounds $\Ncal t(X, r)$ are increasingly difficult to compute, since they are the objective values of infinite-dimensional conic programs whose number of constraints and dimension of underlying cones grow accordingly to $t$. We show that these programs satisfy strong duality and derive a finite dimensional semidefinite program to approximate $\Ncal 2(S 2, r)$ to arbitrary precision. Our results rely in part on the moment methods developed by de Laat a
