Boolean Satisfiability (SAT) is one of the problems at the core of theoretical computer science. It was the first problem proven to be NP-complete by Cook and, independently, by Levin. Nowadays it is conjectured that SAT cannot be solved in sub-exponential time. Thus, it is generally assumed that SAT and its restricted version k-SAT are hard to solve. However, state-of-the-art SAT solvers can solve even huge practical instances of these problems in a reasonable amount of time. Why is SAT hard in theory, but easy in practice? One approach to answering this question is investigating the average runtime of SAT. In order to analyze this average runtime the random k-SAT model was introduced. The model generates all k-SAT instances with n variables and m clauses with uniform probability. Researching random k-SAT led to a multitude of insights and tools for analyzing random structures in general. One major observation was the emergence of the so-called satisfiability threshold: A phase transition point in the number of clauses at which the generated formulas go from asymptotically almost surely satisfiable to asymptotically almost surely unsatisfiable. Additionally, instances around the threshold seem to be particularly hard to solve. In this thesis we analyze a more general model of random k-SAT that we call non-uniform random k-SAT. In contrast to the classical model each of the n Boolean variables now has a distinct probability of being drawn. For each of the m clauses we draw k variables according to the variable distribution and choose their signs uniformly at random. Non-uniform random k-SAT gives us more control over the distribution of Boolean variables in the resulting formulas. This allows us to tailor distributions to the ones observed in practice. Notably, non-uniform random k-SAT contains the previously proposed models random k-SAT, power-law random k-SAT and geometric random k-SAT as special cases. We analyze the satisfiability threshold in non-uniform random k-SAT depending on the variable probability distribution. Our goal is to derive conditions on this distribution under which an equivalent of the satisfiability threshold conjecture holds. We start with the arguably simpler case of non-uniform random 2-SAT. For this model we show under which conditions a threshold exists, if it is sharp or coarse, and what the leading constant of the threshold function is. These are exactly the three ingredients one needs in order to prove or disprove the satisfiability threshold conjecture. For non-uniform random k-SAT with k=3 we only prove sufficient conditions under which a threshold exists. We also show some properties of the variable probabilities under which the threshold is sharp in this case. These are the first results on the threshold behavior of non-uniform random k-SAT.

Background Reproducible benchmarking is important for assessing the effectiveness of novel feature selection approaches applied on gene expression data, especially for prior knowledge approaches that incorporate biological information from online knowledge bases. However, no full-fledged benchmarking system exists that is extensible, provides built-in feature selection approaches, and a comprehensive result assessment encompassing classification performance, robustness, and biological relevance. Moreover, the particular needs of prior knowledge feature selection approaches, i.e. uniform access to knowledge bases, are not addressed. As a consequence, prior knowledge approaches are not evaluated amongst each other, leaving open questions regarding their effectiveness. Results We present the Comprior benchmark tool, which facilitates the rapid development and effortless benchmarking of feature selection approaches, with a special focus on prior knowledge approaches. Comprior is extensible by custom approaches, offers built-in standard feature selection approaches, enables uniform access to multiple knowledge bases, and provides a customizable evaluation infrastructure to compare multiple feature selection approaches regarding their classification performance, robustness, runtime, and biological relevance. Conclusion Comprior allows reproducible benchmarking especially of prior knowledge approaches, which facilitates their applicability and for the first time enables a comprehensive assessment of their effectiveness

In this paper, we analyze stochastic dynamic pricing and advertising differential games in special oligopoly markets with constant price and advertising elasticity. We consider the sale of perishable as well as durable goods and include adoption effects in the demand. Based on a unique stochastic feedback Nash equilibrium, we derive closed-form solution formulas of the value functions and the optimal feedback policies of all competing firms. Efficient simulation techniques are used to evaluate optimally controlled sales processes over time. This way, the evolution of optimal controls as well as the firms’ profit distributions are analyzed. Moreover, we are able to compare feedback solutions of the stochastic model with its deterministic counterpart. We show that the market power of the competing firms is exactly the same as in the deterministic version of the model. Further, we discover two fundamental effects that determine the relation between both models. First, the volatility in demand results in a decline of expected profits compared to the deterministic model. Second, we find that saturation effects in demand have an opposite character. We show that the second effect can be strong enough to either exactly balance or even overcompensate the first one. As a result we are able to identify cases in which feedback solutions of the deterministic model provide useful approximations of solutions of the stochastic model.

These days design thinking is no longer a “new approach”. Among practitioners, as well as academics, interest in the topic has gathered pace over the last two decades. However, opinions are divided over the longevity of the phenomenon: whether design thinking is merely “old wine in new bottles,” a passing trend, or still evolving as it is being spread to an increasing number of organizations and industries. Despite its growing relevance and the diffusion of design thinking, knowledge on the actual status quo in organizations remains scarce. With a new study, the research team of Prof. Uebernickel and Stefanie Gerken investigates temporal developments and changes in design thinking practices in organizations over the past six years comparing the results of the 2015 “Parts without a whole” study with current practices and future developments. Companies of all sizes and from different parts of the world participated in the survey. The findings from qualitative interviews with experts, i.e., people who have years of knowledge with design thinking, were cross-checked with the results from an exploratory analysis of the survey data. This analysis uncovers significant variances and similarities in how design thinking is interpreted and applied in businesses.

Business process simulation is an important means for quantitative analysis of a business process and to compare different process alternatives. With the Business Process Model and Notation (BPMN) being the state-of-the-art language for the graphical representation of business processes, many existing process simulators support already the simulation of BPMN diagrams. However, they do not provide well-defined interfaces to integrate new concepts in the simulation environment. In this work, we present the design and architecture of a proof-of-concept implementation of an open and extensible BPMN process simulator. It also supports the simulation of multiple BPMN processes at a time and relies on the building blocks of the well-founded discrete event simulation. The extensibility is assured by a plug-in concept. Its feasibility is demonstrated by extensions supporting new BPMN concepts, such as the simulation of business rule activities referencing decision models and batch activities.