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Helly theorems and generalized linear programming

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Nina AmentaAuthors Info & Claims

SCG '93: Proceedings of the ninth annual symposium on Computational geometry

Pages 63 - 72

https://doi.org/10.1145/160985.161000

Published: 01 July 1993 Publication History

Abstract

Recent combinatorial algorithms for linear programming also solve certain non-linear problems. We call these Generalized Linear Programming, or GLP, problems. One way in which convexity has been generalized by mathematicians is through a collection of results called the Helly theorems. We show that the every GLP problem implies a Helly theorem, and we give two paradigms for constructing a GLP problem from a Helly theorem. We give many applications, including linear expected time algorithms for finding line transversals and hyperplane fitting in convex metrics. These include GLP problems with the surprising property that the constraints are non-convex or even disconnected. We show that some Helly theorems cannot be turned into GLP problems.

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Cited By

Brandenberg RKönig S(2013)No Dimension-Independent Core-Sets for Containment Under HomotheticsDiscrete & Computational Geometry10.1007/s00454-012-9462-049:1(3-21)Online publication date: 1-Jan-2013
https://dl.acm.org/doi/10.1007/s00454-012-9462-0
Demouth JDevillers OGlisse MGoaoc X(2009)Helly-Type Theorems for Approximate CoveringDiscrete & Computational Geometry10.1007/s00454-009-9167-142:3(379-398)Online publication date: 21-Apr-2009
https://doi.org/10.1007/s00454-009-9167-1
Gärtner BMatoušek JRüst LŠkovroň P(2008)Violator spacesDiscrete Applied Mathematics10.1016/j.dam.2007.08.048156:11(2124-2141)Online publication date: 1-Jun-2008
https://dl.acm.org/doi/10.1016/j.dam.2007.08.048
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Index Terms

Helly theorems and generalized linear programming
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '93: Proceedings of the ninth annual symposium on Computational geometry

July 1993

406 pages

ISBN:0897915828

DOI:10.1145/160985

Chairman:
Chee Yap

Copyright © 1993 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 July 1993

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9SCG93

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9SCG93: Ninth Symposium on Computational Geometry

May 18 - 21, 1993

California, San Diego, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Brandenberg RKönig S(2013)No Dimension-Independent Core-Sets for Containment Under HomotheticsDiscrete & Computational Geometry10.1007/s00454-012-9462-049:1(3-21)Online publication date: 1-Jan-2013
https://dl.acm.org/doi/10.1007/s00454-012-9462-0
Demouth JDevillers OGlisse MGoaoc X(2009)Helly-Type Theorems for Approximate CoveringDiscrete & Computational Geometry10.1007/s00454-009-9167-142:3(379-398)Online publication date: 21-Apr-2009
https://doi.org/10.1007/s00454-009-9167-1
Gärtner BMatoušek JRüst LŠkovroň P(2008)Violator spacesDiscrete Applied Mathematics10.1016/j.dam.2007.08.048156:11(2124-2141)Online publication date: 1-Jun-2008
https://dl.acm.org/doi/10.1016/j.dam.2007.08.048
Puerto JRodríguez-Chía ATamir A(2008)On the Planar Piecewise Quadratic 1-Center ProblemAlgorithmica10.1007/s00453-008-9210-257:2(252-283)Online publication date: 19-Jul-2008
https://doi.org/10.1007/s00453-008-9210-2
Li Yi Jacobs D(2007)Efficiently Determining Silhouette Consistency2007 IEEE Conference on Computer Vision and Pattern Recognition10.1109/CVPR.2007.383161(1-8)Online publication date: Jun-2007
https://doi.org/10.1109/CVPR.2007.383161
Dourado MProtti FSzwarcfiter J(2006)Computational aspects of the Helly property: a surveyJournal of the Brazilian Computer Society10.1007/BF0319238512:1(7-33)Online publication date: Feb-2006
https://doi.org/10.1007/BF03192385
Eppstein DSleator D(1994)Average case analysis of dynamic geometric optimizationProceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/314464.314481(77-86)Online publication date: 23-Jan-1994
https://dl.acm.org/doi/10.5555/314464.314481
Amenta NMehlhorn K(1994)Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theoremProceedings of the tenth annual symposium on Computational geometry10.1145/177424.178064(340-347)Online publication date: 10-Jun-1994
https://dl.acm.org/doi/10.1145/177424.178064
Matoušek JMehlhorn K(1994)On geometric optimization with few violated constraintsProceedings of the tenth annual symposium on Computational geometry10.1145/177424.178039(312-321)Online publication date: 10-Jun-1994
https://dl.acm.org/doi/10.1145/177424.178039

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