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Titel | Smoothed Analysis of Selected Optimization Problems and Algorithms |
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Autor | Tobias Brunsch |

Publikationsform | Dissertation |

Abstract | Optimization problems arise in almost every eld of economics, engineering, and science. Many of these problems are well-understood in theory and sophisticated algorithms exist to solve them efficiently in practice. Unfortunately, in many cases the theoretically most efficient algorithms perform poorly in practice. On the other hand, some algorithms are much faster than theory predicts. This discrepancy is a consequence of the pessimism inherent in the framework of worst-case analysis, the predominant analysis concept in theoretical computer science. We study selected optimization problems and algorithms in the framework of smoothed analysis in order to narrow the gap between theory and practice. In smoothed analysis, an adversary specifies the input, which is subsequently slightly perturbed at random. As one example we consider the successive shortest path algorithm for the minimumcost flow problem. While in the worst case the successive shortest path algorithm takes exponentially many steps to compute a minimum-cost flow, we show that its running time is polynomial in the smoothed setting. Another problem studied in this thesis is makespan minimization for scheduling with related machines. It seems to be unlikely that there exist fast algorithms to solve this problem exactly. This is why we consider three approximation algorithms: the jump algorithm, the lex-jump algorithm, and the list scheduling algorithm. In the worst case, the approximation guarantees of these algorithms depend on the number of machines. We show that there is no such dependence in smoothed analysis. We also apply smoothed analysis to multicriteria optimization problems. In particular, we consider integer optimization problems with several linear objectives that have to be simultaneously minimized. We derive a polynomial upper bound for the size of the set of Pareto-optimal solutions contrasting the exponential worst-case lower bound. As the icing on the cake we find that the insights gained from our smoothed analysis of the running time of the successive shortest path algorithm lead to the design of a randomized algorithm for finding short paths between two given vertices of a polyhedron. We see this result as an indication that, in future, smoothed analysis might also result in the development of fast algorithms. |

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