Complete enumeration of small realizable oriented matroids

TL;DR

This paper develops algorithms to classify small oriented matroids by realizability, enabling complete enumeration of certain low-dimensional configurations and polytopes, despite the NP-hardness of the problem.

Contribution

It introduces algorithmic methods for classifying realizable oriented matroids and provides complete enumerations for specific small configurations and polytopes.

Findings

All combinatorial types of 3D configurations of 8 points identified

All 2D configurations of 9 points classified

All 5D configurations of 9 points and 5-polytopes with 9 vertices enumerated

Abstract

Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points and 5-dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices.