We consider the asymmetric traveling salesperson problem with #-parameterized triangle inequality for # 2 , 1). That means, the edge weights fulfill w(u, v) (w(u, x) + w(x, v)) for all nodes u, v, x. Chandran and Ram [8] recently gave the first constant factor approximation algorithm with polynomial running time for this problem. They achieve performance ratio . We devise an approximation algorithm with performance 3 , which is better than the one of Chandran and Ram for # [0.5437, 1), that is, for the particularly interesting large values of #.

Abstract. We consider the following reoptimization scenario: Given an instance of an optimization problem together with an optimal solution, we want to find a high-quality solution for a locally modified instance. The naturally arising question is whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. In this paper, we survey some partial answers to this questions: Using some variants of the traveling salesman problem and the Steiner tree problem as examples, we show that the answer to this question depends on the considered problem and the type of local modification and can be totally different: For instance, for some reoptimization variant of the metric TSP, we get a 1.4-approximation improving on the best known approximation ratio of 1.5 for the classical metric TSP. For the Steiner tree problem on graphs with bounded cost function, which is APX-hard in its classical formulation, we even obtain a PTAS for the reoptimization variant. On the other hand, for a variant of TSP, where some vertices have to be visited before a prescribed deadline, we are able to show that the reoptimization problem is exactly as hard to approximate as the original problem.

In this paper we study the problem of determining threedimensional orientations for noisy projections of randomly oriented identical particles. The problem is of central importance in the tomographic reconstruction of the density map of macromolecular complexes from electron microscope images and it has been studied intensively for more than 30 years.

In this report we study the problem of determining three-dimensional orientations for noisy projections of randomly oriented identical particles. The problem is of central importance in the tomographic reconstruction of the density map of macromolecular complexes from electron microscope images and it has been studied intensively for more than 30 years.

2 Scientific Work

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