Maximilian Witek

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– We define approximative solution notions and investigate in which cases polynomial-time solvability translates from one to another notion. Moreover, for problems where all objectives have to be minimized, approximability results translate from single-objective to multiobjective optimization such that the relative error degrades only by a constant factor.(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 satisfiabilty problem is 1/2-approximable. 1998 ACM Subject(More)
We systematically study the hardness and the approximability of combinatorial multiobjective NP optimization problems (multi-objective problems, for short). We define solution notions that precisely capture the typical algorithmic tasks in multiobjective optimization. These notions inherit polynomial-time Turing reducibility from multivalued functions,(More)
We improve and derandomize the best known approximation algorithm for the twocriteria metric traveling salesman problem (2-TSP). More precisely, we construct a deterministic 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 the(More)
Instances of optimization problems with multiple objectives can have several optimal solutions whose cost vectors are incomparable. This ambiguity leads to several reasonable notions for solving multiobjective problems. Each such notion defines a class of multivalued functions. We systematically investigate the computational complexity of these classes.(More)
We investigate the autoreducibility and mitoticity of complete sets for several classes with respect to different polynomial-time and logarithmic-space reducibility notions. Previous work in this area focused on polynomial-time reducibility notions. Here we obtain new mitoticity and autoreducibility results for the classes EXP and NEXP with respect to some(More)
For every list of integers x1, . . . , xm there is some j such that x1+ · · ·+xj −xj+1−· · ·−xm ≈ 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the xi are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now(More)
We propose a generalized definition for the multi-objective traveling salesman problem which uses multigraphs and which allows multiple visits of cities. The definition has two benefits: it captures typical real-world scenarios and it contains the conventional definition (componentwise metric cost function) as a special case. We provide approximation(More)
We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain: • For NP and all other classes of the PH: each ≤ m -complete set is ≤ log T -autoreducible. • For P, ∆pk, NEXP: each ≤ m -complete set is a disjoint union of two ≤ log 2-tt-complete(More)