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In Part I: "The flow-decomposition problem", we introduce and discuss the flow-decomposition problem. Given a flow F, this problem consists of decomposing the flow into a set of paths optimizing specific properties of those paths. We introduce different types of decompositions, such as integer decompositions and alpha-decompositions, and provide two formulations of the set of feasible decompositions.
We show that the problem of minimizing the longest path in a decomposition is NP-hard, even for fractional solutions. Then we develop an algorithm based on column generation which is able to solve the problem.
Tight upper bounds on the optimal objective value help to improve the performance.
To find upper bounds on the optimal solution for the shortest longest path problem, we develop several heuristics and analyze their quality. On pearl graphs we prove a constant approximation ratio of 2 and 3 respectively for all heuristics. A numerical study on random pearl graphs shows that the solutions generated by the heuristics are usually much better than this worst-case bound.
In Part II: "Construction and analysis of evacuation models using flows over time", we consider two optimization models in the context of evacuation planning. The first model is a parameter-based quickest flow model with time-dependent supply values. We give a detailed description of the network construction and of how different scenarios are modeled by scenario parameters. In a second step we analyze the effect of the scenario parameters on the evacuation time. Understanding how the different parameters influence the evacuation time allows us to provide better advice for evacuation planning and allows us to predict evacuation times without solving additional optimization problems. To understand the effect of the time-dependent supply values, we consider the quickest path problem with time-dependent supply values and provide a solution algorithm. The results from this consideration are generalized to approximate the behavior of the evacuation times in the context of quickest flow problems.
The second model we consider is a path-based model for evacuation in the presence of a dynamic cost function. We discuss the challenges of this model and provide ideas for how to approach the problem from different angles. We relate the problem to the flow-decomposition problem and consider the computation of evacuation paths with dynamic costs for large capacities. For the latter method we provide heuristics to find paths and compare them to the optimal solutions by applying the methods to two evacuation scenarios. An analysis shows that the paths generated by the heuristic yield close to optimal solutions and in addition have several desirable properties for evacuation paths which are not given for the optimal solution.

The formulation of the decoding problem for linear block codes as an integer program (IP) with a rather tight linear programming (LP) relaxation has made a central part of channel coding accessible for the theory and methods of mathematical optimization, especially integer programming, polyhedral combinatorics and also algorithmic graph theory, since the important class of turbo codes exhibits an inherent graphical structure. We present several novel models, algorithms and theoretical results for error-correction decoding based on mathematical optimization. Our contribution includes a partly combinatorial LP decoder for turbo codes, a fast branch-and-cut algorithm for maximum-likelihood (ML) decoding of arbitrary binary linear codes, a theoretical analysis of the LP decoder's performance for 3-dimensional turbo codes, compact IP models for various heuristic algorithms as well as ML decoding in combination with higher-order modulation, and, finally, first steps towards an implementation of the LP decoder in specialized hardware. The scientific contributions are presented in the form of seven revised reprints of papers that appeared in peer-reviewed international journals or conference proceedings. They are accompanied by an extensive introductory part that reviews the basics of mathematical optimization, coding theory, and the previous results on LP decoding that we rely on afterwards.

Mathematical models of species dispersal and the resilience of metapopulations against habitat loss
(2021)

Habitat loss and fragmentation due to climate and land-use change are among the biggest threats to biodiversity, as the survival of species relies on suitable habitat area and the possibility to disperse between different patches of habitat. To predict and mitigate the effects of habitat loss, a better understanding of species dispersal is needed. Graph theory provides powerful tools to model metapopulations in changing landscapes with the help of habitat networks, where nodes represent habitat patches and links indicate the possible dispersal pathways between patches.
This thesis adapts tools from graph theory and optimisation to study species dispersal on habitat networks as well as the structure of habitat networks and the effects of habitat loss. In chapter 1, I will give an introduction to the thesis and the different topics presented in this thesis. Chapter 2 will then give a brief summary of tools used in the thesis.
In chapter 3, I present our model on possible range shifts for a generic species. Based on a graph-based dispersal model for a generic aquatic invertebrate with a terrestrial life stage, we developed an optimisation model that models dispersal directed to predefined habitat patches and yields a minimum time until these patches are colonised with respect to the given landscape structure and species dispersal capabilities. We created a time-expanded network based on the original habitat network and solved a mixed integer program to obtain the minimum colonisation time. The results provide maximum possible range shifts, and can be used to estimate how fast newly formed habitat patches can be colonised. Although being specific for this simulation model, the general idea of deriving a surrogate can in principle be adapted to other simulation models.
Next, in chapter 4, I present our model to evaluate the robustness of metapopulations. Based on a variety of habitat networks and different generic species characterised by their dispersal traits and habitat demands, we modeled the permanent loss of habitat patches and subsequent metapopulation dynamics. The results show that species with short dispersal ranges and high local-extinction risks are particularly vulnerable to the loss of habitat across all types of networks. On this basis, we then investigated how well different graph-theoretic metrics of habitat networks can serve as indicators of metapopulation robustness against habitat loss. We identified the clustering coefficient of a network as the only good proxy for metapopulation robustness across all types of species, networks, and habitat loss scenarios.
Finally, in chapter 5, I utilise the results obtained in chapter 4 to identify the areas in a network that should be improved in terms of restoration to maximise the metapopulation robustness under limited resources. More specifically, we exploit our findings that a networkâ€™s clustering coefficient is a good indicator for metapopulation robustness and develop two heuristics, a Greedy algorithm and a deducted Lazy Greedy algorithm, that aim at maximising the clustering coefficient of a network. Both algorithms can be applied to any network and are not specific to habitat networks only.
In chapter 6, I will summarize the main findings of this thesis, discuss their limitations and give an outlook of future research topics.
Overall this thesis develops frameworks to study the behaviour of habitat networks and introduces mathematical tools to ecology and thus narrows the gap between mathematics and ecology. While all models in this thesis were developed with a focus on aquatic invertebrates, they can easily be adapted to other metapopulations.