Combinatorial face enumeration in arrangements and oriented matroids

Tropical Hyperplane Arrangements and Oriented Matroids

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.

Complete enumeration of small realizable oriented matroids

Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, are often more important than their metric structures. For example, the convexity, the face lattice of the convex hull and all possible triangulations of a given point configurat...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Combinatorial and Geometric Approaches to Counting Problems on Linear Matroids, Graphic Arrangements, and Partial Orders

For typical #P-hard problems on graphs, we have recently proposed an approach to solve those problems of moderate size rigorously by means of the binary decision diagram, BDD [12, 13]. This paper extends this approach to counting problems on linear matroids, graphic arrangements and partial orders, most of which are already known to be #P-hard, with using geometric properties. E cient algorithm...

Fully Optimal Bases and the Active Bijection in Graphs, Hyperplane Arrangements, and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bijective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterizatio...