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 R 3 and that of Motzkin on ≤12 lines in R 2 , 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.

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 5dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices.

An acyclic USO on a hypercube is formed by directing its edges in such as way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modeled as pivoting in a unit hypercube of the same dimension with an abstract objective function, and vice versa. In such a way, Zadeh's 'least entered rule' and other history based pivot rules can be applied to the problem of finding the global sink of an acyclic USO. In this paper we present some theoretical and empirical results on the existence of acyclic USOs for which the various history based pivot rules can be made to follow a Hamiltonian path. In particular, we develop an algorithm that can enumerate all such paths up to dimension 6 using efficient pruning techniques. We show that Zadeh's original rule admits Hamiltonian paths up to dimension 9 at least, and prove that most of the other rules do not for all dimensions greater than 5. √ d) [7]. A non-trivial upper bound on Zadeh's rule is still unknown.

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