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1 PRISME - Geometry, Algorithms and Robotics CRISAM - Inria Sophia Antipolis - Méditerranée

Abstract : A set of objects is $k$-pierceable if there exists a set of $k$ points such that each object is pierced by contains at least one of these points. Finding the smallest integer $k$ such that a set is $k$-pierceable is NP-complete. In this technical report, we present efficient algorithms for finding a piercing set i.e., a set of $k$ points as above for several classes of convex objects and small values of $k$. In some of the cases, our algorithms imply known as well as new Helly-type theorems, thus adding to previous results of Danzer and Grünbaum who studied the case of axis-parallel boxes. The problems studied here are related to the collection of optimization problems in which one seeks the smallest scaling factor of a centrally symmetric convex object $K$, so that a set of points can be covered by $k$ congruent homothets of $K$.


Author: Matthew J. Katz - Franck Nielsen

Source: https://hal.archives-ouvertes.fr/


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