Complementary Vertices and Adjacency Testing in Polytopes

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.