Unique Sink Orientations of Cubes

Subject of this thesis is the theory of unique sink orientations of cubes. Such orientations are suitable to model problems from different areas of combinatorial optimization. In particular, unique sink orientations are closely related to the running time of the simplex algorithm. In the following, we try to answer three main questions: How can optimization problems be translated into the framework of unique sink orientations? What structural properties do unique sink orientations have? And how difficult is the algorithmic problem of finding the sink of a unique sink orientation? In connection to the first question, the main result is a reduction from linear programming to unique sink orientations. Although the connection to linear programming was the core motivation for our studies, it was not clear in the beginning how general linear programs can be fit into the theory of unique sink orientations. The reduction presented in this thesis closes this gap. For the second question we can provide several construction schemes for unique sink orientations. On the one hand we know schemes which allow us to construct all unique sink orientations. On the other hand we present easier constructions, which are still powerful enough to provide us with a number of interesting orientations. The hope that unique sink orientations on their own carry an interesting algebraic structure turns out to be wrong. Equipped with the construction schemes just mentioned we are able to give some answers to the third question about the algorithmic complexity. The true complexity of the problem of finding a sink of a unique sink orientation remains open. But we can provide first lower bounds for special algorithms as well as for the general case. Furthermore, it turns out that the algorithmic problem is NP-hard only if NP=coNP.