Complementary vertices and adjacency testing in polytopes

TL;DR

This paper presents a new theoretical result about the existence of multiple complementary vertex pairs in simple polytopes and introduces an efficient adjacency testing method that improves upon previous algorithms, applicable to both simple and non-simple polytopes.

Contribution

It proves that simple polytopes with a complementary vertex pair have at least two such pairs and develops an O(n) adjacency test that is exact for simple polytopes and a filtering tool for non-simple ones.

Findings

Theoretical proof of multiple complementary vertex pairs in simple polytopes.

An O(n) adjacency test for vertices in polytopes.

Improved adjacency testing algorithms over previous methods.

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 at least two such pairs, which can be chosen to be disjoint. Using this result, we improve adjacency testing for vertices in both simple and non-simple polytopes: given a polytope in the standard form {x \in R^n | Ax = b and 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 + n V^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.