Maximilian Witek

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We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1 /2-approximable on directed graphs and randomized 2 /3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiabilty problem is 1 /2-approximable. 1 Introduction We(More)
We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are: – We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions. – These solution notions induce corresponding NP-hardness notions for(More)
We improve and derandomize the best known approximation algorithm for the two-criteria metric traveling salesman problem (2-TSP). More precisely, we construct a deter-ministic 2-approximation which answers an open question by Manthey. Moreover, we show that 2-TSP is randomized (3 /2 + ε, 2)-approximable, and we give the first randomized approximations for(More)
We systematically study the hardness and the approximability of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). We define solution notions that precisely capture the typical algorithmic tasks in multi-objective optimization. These notions inherit polynomial-time Turing reducibility from mul-tivalued functions,(More)
For every list of integers x 1 ,. .. , x m there is some j such that x 1 + · · · + x j − x j+1 − · · · − x m ≈ 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still(More)
Acknowledgements I am grateful for all the support I received during my studies and the completion of this thesis. Most of all, I would like to thank my advisor Christian Glaßer for his continuous support and for sharing his profound understanding of theory and mathematics with me. Special thanks go to Heinz Schmitz, who encouraged my interest in the(More)