, Volume 12, Issue 1, pp 241-261

Helly-type theorems and Generalized Linear Programming

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Abstract

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.

The author was supported at Berkeley by a University of California President’s Dissertation Year Fellowship. Some of this work was done while visiting the Freie Universität, Berlin.