Phylogenetic diversity (PD) plays an important role in biodiversity, conservation, and evolutionary studies by measuring the diversity of a set of taxa based on their phylogenetic relationships. In phylogenetic trees, a subset of k taxa with maximum PD can be found by a simple and efficient greedy algorithm. However, this algorithmic tractability is lost when considering phylogenetic networks, which incorporate reticulate evolutionary events such as hybridization and horizontal gene transfer. To address this challenge, we introduce PaNDA (Phylogenetic Network Diversity Algorithms), the first software package and interactive graphical user-interface for exploring, visualizing, and maximizing diversity in phylogenetic networks. PaNDA includes a novel algorithm to find a subset of k taxa with maximum diversity, running in polynomial time for networks of bounded scanwidth, a measure of tree-likeness of a network that grows slower than the well-known level measure. This algorithm considers the variant of PD on networks in which the branch lengths of all paths from the root to the selected taxa contribute towards their diversity. We demonstrate the scalability of this algorithm on simulated networks, successfully analyzing level-15 networks with up to 200 taxa in seconds. We also provide a proof-of-concept analysis using a phylogenetic network on Xiphophorus species, illustrating how the tool can support diversity studies based on real genomic data. The software is easily installable and freely available at https://github.com/nholtgrefe/panda. Additionally, we extend the definition of PD to semi-directed phylogenetic networks, which are mixed graphs increasingly used in phylogenetic analysis to model uncertainty of the root location. We prove that finding a subset of k taxa with maximum diversity remains NP-hard on semi-directed networks, but do present a polynomial-time algorithm for networks with bounded level. ... Read more

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. ... Read more

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. ... Read more