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arXiv · 1103.2575

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

David Eppstein·Maarten Löffler
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Abstract

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs.

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David Eppstein, Maarten Löffler. 2011-03-14. Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension. https://doi.org/10.1007/s00454-013-9503-3

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