Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces
Mathematical Programming: Series A and B
On the complexity of some basic problems in computational convexity: I.: containment problems
Discrete Mathematics - Special issue: trends in discrete mathematics
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The 2-center problem with obstacles
Journal of Algorithms
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Fast and Robust Smallest Enclosing Balls
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Approximate minimum enclosing balls in high dimensions using core-sets
Journal of Experimental Algorithmics (JEA)
Efficient Algorithms for the Smallest Enclosing Ball Problem
Computational Optimization and Applications
New Algorithms for k-Center and Extensions
COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
No dimension independent core-sets for containment under homothetics
Proceedings of the twenty-seventh annual symposium on Computational geometry
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Minimal containment problems arise in a variety of applications, such as shape fitting and packing problems, data clustering, pattern recognition, or medical surgery. Typical examples are the smallest enclosing ball, cylinder, slab, box, or ellipsoid of a given set of points.Here we focus on one of the most basic problems: minimal containment under homothetics, i.e., covering a point set by a minimally scaled translation of a given container. Besides direct applications this problem is often the base in solving much harder containment problems and therefore fast solution methods are needed, especially in moderate dimensions. While in theory the ellipsoid method suffices to show polynomiality in many cases, extensive studies of implementations exist only for Euclidean containers. Indeed, many applications require more complicated containers.In Plastria (Eur. J. Oper. Res. 29:98---110, 1987) the problem is discussed in a more general setting from the facility location viewpoint and a cutting plane method is suggested. In contrast to Plastria (Eur. J. Oper. Res. 29:98---110, 1987), our approach relies on more and more accurate approximations of the container. For facet and vertex presented polytopal containers the problem can be formulated as an LP, and for many general containers as an SOCP. The experimental section of the paper compares those formulations to the cutting plane method, showing that it outperforms the LP formulations for vertex presented containers and the SOCP formulation for some problem instances.