Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$

Parameterized Approaches for Large-Scale Optimization Problems

In this dissertation, we study challenging discrete optimization problems from the perspective of parameterized complexity. The usefulness of this type of analysis is twofold. First, it can lead to efficient algorithms for large-scale problem instances. Second, the analysis can provide a rigorous explanation for why challenging problems might appear relatively easy in practice. We illustrate the approach on several different problems, including: the maximum clique problem in sparse graphs; 0-1 programs with many conflicts; and the node-weighted Steiner tree problem with few terminal nodes. We also study polyhedral counterparts to fixed-parameter tractable algorithms. Specifically, we provide fixed-parameter tractable extended formulations for independent set in tree-like graphs and for cardinality-constrained vertex covers

Connectivity Constraints in Network Analysis

This dissertation establishes mathematical foundations of connectivity requirements arising in both abstract and geometric network analysis. Connectivity constraints are ubiquitous in network design and network analysis. Aside from the obvious applications in communication and transportation networks, they have also appeared in forest planning, political distracting, activity detection in video sequences and protein-protein interaction networks. Theoretically, connectivity constraints can be analyzed via polyhedral methods, in which we investigate the structure of (vertex)-connected subgraph polytope (CSP). One focus of this dissertation is on performing an extensive study of facets of CSP. We present the first systematic study of non-trivial facets of CSP. One advantage to study facets is that a facet-defining inequality is always among the tightest valid inequalities, so applying facet-defining inequalities when imposing connectivity constraints can guarantee good performance of the algorithm. We adopt lifting techniques to provide a framework to generate a wide class of facet-defining inequalities of CSP. We also derive the necessary and sufficient conditions when a vertex separator inequality, which plays a critical role in connectivity constraints, induces a facet of CSP. Another advantage to study facets is that CSP is uniquely determined by its facets, so full understanding of CSP's facets indicates full understanding of CSP itself. We are able to derive a full description of CSP for a wide class of graphs, including forest and several types of dense graphs, such as graphs with small independence number, s-plex with small s and s-defective cliques with small s. Furthermore, we investigate the relationship between lifting techniques, maximum weight connected subgraph problem and node-weight Steiner tree problem and study the computational complexity of generation of facet-defining inequalities. Another focus of this dissertation is to study connectivity in geometric network analysis. In geometric applications like wireless networks and communication networks, the concept of connectivity can be defined in various ways. In one case, connectivity is imposed by distance, which can be modeled by unit disk graphs (UDG). We create a polytime algorithm to identify large 2-clique in UDG; in another case when connectivity is based on visibility, we provide a generalization of the two-guard problem

De-novo pathway discovery for multi-omics data

In der vorliegenden Arbeit werden Methoden und Software vorgestellt, die es erlauben, aus Hochdurchsatz Omics-Daten und biomolekularen Interaktionsnetzwerken biologisch relevante Muster zu extrahieren. Es wird ein Algorithmus entwickelt, der es ermöglicht, aus großen gerichteten molekularen Interaktionsnetzwerken sog. deregulierte Teilnetzwerke zu extrahieren. Deregulierung wird hierbei über auf die Knoten des Netzwerkes abgebildete Omics-Daten definiert. Es wird eine statistische Grundlage für den vorgestellten Algorithmus diskutiert und eine Evaluierung hinsichtlich methodisch verwandter Verfahren vorgenommen. Der Algorithmus und seine Implementierung, DeRegNet, beruhen auf fraktionaler ganzzahliger Optimierung und erlauben zahlreiche Anwendungsszenarien. Exemplarisch wird die Anwendung auf öffentlich zugängliche Daten des TCGA-Projekts vorgestellt (TCGA: The Cancer Genome Atlas), hier genauer an Hand der Daten zum hepatozellulären Karzinom (Leberkrebs). Weiterhin werden Anwendungen auf eine Studie des Folate One-Carbon Metabolismus im Leberkrebs, als auch auf die phosphoproteomische Regulierung des Saccharomyces cerevisiae (Backhefe) Zellzyklus beschrieben. Abschließend wird auf die allgemeine Architektur und einige Implementationsdetails einer web-basierten API (Application Programming Interface) zur Bereitstellung von DeRegNet eingegangen.This thesis presents algorithms and software which allow the extraction of biologically meaningful patterns from high-throughput multi-omics data and biomolecular networks. It describes the concept and implementation of an algorithm which allows the extraction of deregulated subnetworks from large directed molecular interaction networks based on node scores derived from omics data. Statistical underpinnings of the algorithms are derived and the algorithm is benchmarked against its closest methodological relative. Relying on fractional integer programming, the algorithm and its implementation, DeRegNet, allow many flexible modes of application. I demonstrate the application of the algorithm in the context of the public TCGA (The Cancer Genome Atlas) liver cancer dataset, a study investigating the role of folate one-carbon metabolism in liver cancer and a study about the phosphoproteomic regulation of the Saccharomyces cerevisiae (baker's yeast) cell cycle. Finally, the general architecture and some implementation details of a web-based API for DeRegNet are presented