On piercing sets of objects

Matthew Katz, Franck Nielsen

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations

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 paper, 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 Gruenbaum 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.

Original languageEnglish
Title of host publicationProceedings of the 1996 12th Annual Symposium on Computational Geometry
Pages113-121
Number of pages9
DOIs
StatePublished - 1 Jan 1996
Externally publishedYes
EventProceedings of the 1996 12th Annual Symposium on Computational Geometry - Philadelphia, PA, USA
Duration: 24 May 199626 May 1996

Conference

ConferenceProceedings of the 1996 12th Annual Symposium on Computational Geometry
CityPhiladelphia, PA, USA
Period24/05/9626/05/96

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