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Complementary Vertices and Adjacency Testing in Polytopes

  • Benjamin A. Burton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has a second such pair. Using this result, we improve adjacency testing for vertices in both simple and non-simple polytopes: given a polytope in the standard form \(\{\mathbf{x}\in\mathbb{R}^n\,|\,A\mathbf{x}=\mathbf{b}\ \mbox{and}\ \mathbf{x}\geq 0\}\) and a list of its V vertices, we describe an O(n) test to identify whether any two given vertices are adjacent. For simple polytopes this test is perfect; for non-simple polytopes it may be indeterminate, and instead acts as a filter to identify non-adjacent pairs. Our test requires an O(n 2 V + nV 2) precomputation, which is acceptable in settings such as all-pairs adjacency testing. These results improve upon the more general O(nV) combinatorial and O(n 3) algebraic adjacency tests from the literature.

Keywords

polytopes complementary vertices disjoint facets adjacent vertices vertex enumeration double description method 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Benjamin A. Burton
    • 1
  1. 1.School of Mathematics and PhysicsThe University of QueenslandBrisbaneAustralia

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