2012/04/03 by Komei Fukuda, Fukuda, Komei, Hiroyuki Miyata +3 · 1 citation
Computer Science · Engineering · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Constraint Satisfaction and Optimization #FOS: Mathematics #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.1204.0645
openalex publication_date 2012/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.