Jan Remy

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The MINIMUM WEIGHT TRIANGULATION problem is to find a triangulation T* of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1979. Very recently the problem was shown to be NP-hard by(More)
This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniform distributed over [0, 1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it.(More)
In this paper we introduce a new technique for approximation schemes for geometrical optimization problems. As an example problem, we consider the following variant of the geometric Steiner tree problem. Every point u which is not included in the tree costs a penalty of π(u) units. Furthermore, every Steiner point that we use costs cS units. The goal is to(More)
The (Euclidean) Vehicle Routing Allocation Problem (VRAP) is a generalization of Euclidean TSP. We do not require that all points lie on the salesman tour. However, points that do not lie on the tour are allocated, i.e., they are directly connected to the nearest tour point, paying a higher (per-unit) cost. More formally, the input is a set of points P ⊂ R(More)
Given a set of points P in the plane and profits (or prizes) π : P → R ≥0 we want to select a maximum profit set X ⊆ P which maximizes P p∈X π(p) − µ(X) for some particular criterion µ(X). In this paper we consider four such criteria, namely the perimeter and the area of the smallest axis-parallel rectangle containing X, and the perimeter and the area of(More)
This thesis is devoted to geometric optimization problems of the following kind: given a set of points P ⊂ R d , we wish to compute a certain straight-line graph on P with minimum length, such as a shortest Steiner tree, a salesman tour or a triangulation. As most of these problems are N P-hard and many of them even in the strong sense, approximation(More)
Preface I am deeply grateful to all which have contributed to this work by their comments and discussions, by hints, advice, and patience. In particular, I would like to thank For allowing me to present joint work here, many thanks are due to my coauthors and colleagues The algorithms group (" LEA ") at the Technische Universität München has always provided(More)