In evolutionary biology, phylogenetic networks are graphs that provide a flexible framework for representing complex evolutionary histories that involve reticulate evolutionary events. Recently, phylogenetic studies have started to focus on a special class of such networks called semi-directed networks. These graphs are defined as mixed graphs that can be obtained by de-orienting some of the arcs in some rooted phylogenetic network, that is, a directed acyclic graph whose leaves correspond to a collection of species and that has a single source or root vertex. However, this definition of semi-directed networks is implicit in nature since it is not clear when a mixed-graph enjoys this property or not. In this paper, we introduce novel, explicit mathematical characterizations of semi-directed networks, and also multi-semi-directed networks, that is mixed graphs that can be obtained from directed phylogenetic networks that may have more than one root. In addition, through extending foundational tools from the theory of rooted networks into the semi-directed setting—such as cherry picking sequences, omnians, and path partitions—we characterize when a (multi-)semi-directed network can be obtained by de-orienting some rooted network that is contained in one of the well-known classes of tree-child, orchard, tree-based or forest-based networks. These results address structural aspects of (multi-)semi-directed networks and pave the way to improved theoretical and computational analyses of such networks, for example, within the development of algebraic evolutionary models that are based on such networks.

Phylogenetic networks are a specific type of directed acyclic graph (DAG), used to depict evolutionary relationships among, for example, species or other groups of organisms. To solve computationally hard problems, treewidth has been used to parametrize algorithms in phylogenetics. In the hope of simplifying the algorithmic design process, Berry, Scornavacca and Weller recently proposed a new measure of tree-likeness that takes into account the directions of the arcs: scanwidth. They showed that the corresponding decision problem of this parameter - which can be seen as a variant of directed cutwidth, using a tree instead of a linear ordering - is NP-complete. This thesis aims to widen the structural knowledge of scanwidth and to find efficient ways of computing it on general DAGs, both by exact and heuristic algorithms.

With the help of reduction rules, we construct an explicit dynamic programming algorithm that computes scanwidth exactly, along with its corresponding tree extension, in O(k * n^k * m) time for rooted DAGs of scanwidth k. This slicewise polynomial algorithm proves that computing the scanwidth is in the complexity class XP. The algorithm also functions as an FPT algorithm for networks of level-l, with the complexity bounded by O(2^4l-1* l * n + n²). It performs well in practice, being able to compute the scanwidth of networks up to 30 reticulations and 100 leaves within 500 seconds.

On the heuristic side, an algorithm that repeatedly splits at a specific type of smallest cut is proposed. Enhanced with simulated annealing, this heuristic shows promising results, obtaining an average approximation ratio of 1.5 for large synthetic networks of 30 reticulations and 100 leaves. Applied to a real-world dataset of networks, the heuristic performs near-optimal. Although we prove that the scanwidth is always greater than or equal to the treewidth, experiments show that they are close to each other in practice. This further motivates the use of scanwidth over treewidth as a parameter in algorithms.

In this thesis we analyse the class of maximum coverage problems. For all discussed problems, linear programs are formulated. Using the notion of submodularity, we prove that for the weighted version of the basic Maximum Coverage problem, where the weights differ per set, a polynomial-time greedy algorithm guarantees a (1 - 1/e)-approximation of the optimal solution. This improves the already known bound of (1 - 1/e - ε), for all ε > 0. We then show that the same result holds true, if we allow elements to be covered by multiple sets. Furthermore, a completely novel extension is introduced, where weights differ per combination of sets. It is proved that, under the assumption that the weights are submodular and increasing, a greedy algorithm still provides a (1 – 1/e)-approximation. 
The latter algorithm is tested in the framework of optimal sensor selection. To this end, we consider all official weather stations in the Netherlands as our sensor group. We test the performance of the approximation algorithm, if some of the assumptions do not hold and no theoretical bounds exists. Corresponding weights are calculated, using two approaches: inverse distance weighting and multiple linear regression. For both approaches the in practice performance of the greedy algorithm is shown to be even higher than (1 – 1/e), even though not all assumptions hold. Finally, the corresponding selection of weather stations is shown.