Discrete & Computational Geometry

, Volume 12, Issue 3, pp 241–261

Helly-type theorems and Generalized Linear Programming

  • N. Amenta

DOI: 10.1007/BF02574379

Cite this article as:
Amenta, N. Discrete Comput Geom (1994) 12: 241. doi:10.1007/BF02574379


Recent combinatorial algorithms for linear programming can also be applied to certain nonlinear problems. We call these Generalized Linear-Programming, or GLP, problems. We connect this class to a collection of results from combinatorial geometry called Helly-type theorems. We show that there is a Helly-type theorem about the constraint set of every GLP problem. Given a familyH of sets with a Helly-type theorem, we give a paradigm for finding whether the intersection ofH is empty, by formulating the question as a GLP problem. This leads to many applications, including linear expected time algorithms for finding line transversals and mini-max hyperplane fitting. Our applications include GLP problems with the surprising property that the constraints are nonconvex or even disconnected.

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • N. Amenta
    • 1
    • 2
  1. 1.Computer ScienceUniversity of CaliforniaBerkeleyUSA
  2. 2.The Geometry CenterMinneapolisUSA