Systems biology is an interdisciplinary field of research that combines mathematics, computer science, and engineering in order to analyse biological processes. It has become more and more important in the last two decades, in particular because of successful applications for human health and biotechnology. It aims at simulating biological systems as mathematical models to support time- and cost-intensive research in laboratories. To do so, researchers create formalisms, algorithms, and techniques which can be widely used. One technique to obtain data describing biological entities is genome sequencing. Using modern high-throughput sequencing, increasing knowledge is gained about genomes which can then be used in order to reconstruct metabolic processes and networks of the organisms. Success does not come for free and data gathered with modern techniques is often too large to be analysed by hand. Therefore, methods which extract relevant information from data are in great demand. In this thesis, we introduce different methods which reduce given data in metabolic networks in a meaningful way. We present a technique which computes minimal metabolic subnetworks which are still able to satisfy predefined functionalities. We also develop a method to compute minimum sets of elementary flux modes which compose the network, where the size is significantly reduced compared to the whole set of elementary flux modes. Furthermore, we provide procedures that reduce the number of variables in a given problem in order to accelerate (already existing) algorithms by using information given by the data. Moreover, we develop a novel procedure to compute minimal cut sets on a projected network. This enables us to compute minimal cut sets of larger cardinality than before and to analyse larger networks. This projection of metabolic networks also gives rise to other applications such as computing minimal metabolic behaviours. Even though we apply and suit our methods to real metabolic systems, this thesis is focused on the mathematical methods. In order to create and prove the new techniques we make use of (mixed integer) linear optimisation, polyhedral cones, linear algebra, and oriented matroids.
