Sonoko Moriyama

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ3 and that of Motzkin on ≤12 lines in ℝ2, extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.

Complete Enumeration of Small Realizable Oriented Matroids

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 (Discrete Comput Geom 27:117–136, 2002) 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 also determine all possible combinatorial types of 5-polytopes with nine vertices.

Incremental construction properties in dimension two: shellability, extendable shellability and vertex decomposability

We give new examples of shellable, but not extendably shellable two-dimensional simplicial complexes. They include minimal examples that are smaller than those previously known. We also give new examples of shellable, but not vertex decomposable two-dimensional simplicial complexes, including extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability implies the other. We found these examples by enumerating shellable two-dimensional simplicial complexes that are not pseudomanifolds.

Revisiting Hyperbolic Voronoi Diagrams from Theoretical, Applied and Generalized Viewpoints

Voronoi diagram in the hyperbolic space, hyperbolic Voronoi diagrams for short, as well as that with respect to information-geometric divergences has been investigated since mid 1990’s by Onishi et al. This paper revisits the hyperbolic Voronoi diagram from three standpoints, background theory, new applications, and geometric extensions. First, viewed from statistical estimation and information geometry, in the parametric space of normal distributions, the hyperbolic Voronoi diagram is induced by the Fisher metric while the divergence diagram is given by the Kullback-Leibler divergence on a dually flat structure. We show that the hyperbolic Voronoi diagram becomes flat by a linearization similar to one of two flat coordinates in the dually flat space, and it is a part of some power diagram. This gives another proof for the result recently obtained by Nielsen and Nock on the hyperbolic Klein model, completely different linearized model. Our result is interesting in view of the linearization having information geometric interpretations. Second, from the viewpoint of new applications, we discuss the relation between the hyperbolic Voronoi diagram and the greedy embedding in the hyperbolic plane. Kleinberg proved that in the hyperbolic plane the greedy routing is always possible. We point out that results of previous studies about the greedy embedding use a property that any tree is realized as a hyperbolic Delaunay graph easily. Finally, we generalize sites of hyperbolic Voronoi diagrams. Specifically, we consider Voronoi diagrams of geodesic segments in the upper half-plane and propose an algorithm for them.

From Bell Inequalities to Tsirelson's Theorem

The first part of this paper contains an introduction to Bell inequalities and Tsirelson’s theorem for the non-specialist. The next part gives an explicit optimum construction for the “hard” part of Tsirelson’s theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson’s theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.

A lower bound on opaque sets

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.

The Holt-Klee condition for oriented matroids

Holt and Klee have recently shown that every (generic) LP orientation of the graph of a d-polytope satisfies a directed version of the d-connectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any non-HK OM. This leads to a fundamental question as to whether the Holt-Klee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8-element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no non-HK OM in this class.

h-Assignments of simplicial complexes and reverse search

There is currently no efficient algorithm for deciding whether a given simplicial complex is shellable. We propose a practical method that decides shellability of simplicial complexes based on reverse search, which improves an earlier attempt by Moriyama, Nagai and Imai. We also propose to use Macaulays theorem during the search. This works efficiently in high-dimensional cases.

Revisiting hyperbolic voronoi diagrams in two and higher dimensions from theoretical, applied and generalized viewpoints

This paper revisits hyperbolic Voronoi diagrams, which have been investigated since mid 1990s by Onishi et al., from three standpoints, background theory, new applications, and geometric extensions. First, we review two ideas to compute hyperbolic Voronoi diagrams of points. One of them is Onishis method to compute a hyperbolic Voronoi diagram from a Euclidean Voronoi diagram. The other one is a linearization of hyperbolic Voronoi diagrams. We show that a hyperbolic Voronoi diagram of points in the upper half-space model becomes an affine diagram, which is part of a power diagram in the Euclidean space. This gives another proof of a result obtained by Nielsen and Nock on the hyperbolic Klein model. Furthermore, we consider this linearization from the view point of information geometry. In the parametric space of normal distributions, the hyperbolic Voronoi diagram is induced by the Fisher metric while the divergence diagram is given by the Kullback-Leibler divergence on a dually flat structure. We show that the linearization of hyperbolic Voronoi diagrams is similar to one of two flat coordinates in the dually flat space, and our result is interesting in view of the linearization having information-geometric interpretations. Secondly, from the viewpoint of new applications, we discuss the relation between the hyperbolic Voronoi diagram and the greedy embedding in the hyperbolic plane. Kleinberg proved that in the hyperbolic plane the greedy routing is always possible. We point out that results of previous studies about the greedy embedding use a property that any tree is realized as a hyperbolic Delaunay graph easily. Finally, we generalize hyperbolic Voronoi diagrams. We consider hyperbolic Voronoi diagrams of spheres by two measures and hyperbolic Voronoi diagrams of geodesic segments, and propose algorithms for them, whose ideas are similar to those of computing hyperbolic Voronoi diagrams of points.

A note on shellability and acyclic orientations

In this short note we discuss the shellability of (nonpure) simplicial complexes in terms of acyclic orientations of the facet-ridge incidence graphs, which shows that we can decide shellability only from the facet-ridge incidences and the total number of faces the simplicial complex contains.