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

TL;DR

This paper establishes bounds on the complexity of intersections of halfspaces in high-dimensional spaces, especially when the bounded faces have small dimension, providing new insights into their combinatorial structure.

Contribution

It introduces bounds on the number of vertices and bounded faces of such polyhedra based on the maximum bounded face dimension, and offers polynomial-time algorithms for key computations.

Findings

Vertices are bounded by O(n^d)

Bounded faces are bounded by O(n^{d^2})

Algorithms for computing faces run in polynomial time

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.