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)
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 satisfiablilty problem is 1 /2-approximable.