Testing Orientability for Matroids is NP-complete

Matroids and oriented matroids are fundamental objects in combinatorial geometry. While matroids model the behavior of vector configurations over general fields, oriented matroids model the behavior of vector configurations over ordered fields. For every oriented matroid there is a corresponding underlying oriented matroid. This paper addresses the question how difficult it is to algorithmically decide whether on the other hand one can assign an orientation to a given matroid. We will prove that this problem is NP-complete. 1 Matroids and oriented matroids This paper addresses the question of the algorithmic difficulty of testing whether a matroid is orientable. Matroids and oriented matroids form an abstract generalization of the combinatorial properties of arrangements of hyperplanes. While matroids merely encode incidence information, oriented matroids in addition carry information about the relative positions of the hyperplanes. Throughout this paper we will deal only with matroids and oriented matroids of rank 3, which in an affine setup correspond to arrangements of (pseudo) lines. To avoid unnecessary technical difficulties we will restrict all our definitions to the case of rank 3. We start with a few basic notions that will translate our problem into a problem about arrangements of pseudolines with certain prescribed incidence relations. Consider an ordered collection L = (l1, l2, . . . , ln) of n oriented lines in the usual euclidean plane IR, indexed by the finite index set E = {1, 2, . . . , n}. The lines partition the plane into a cell complex that consists of full-dimensional cells (the so called topes of the arrangement), of one-dimensional cells (line segments and rays), and of zero-dimensional cells (the vertices of the arrangement). In a canonical way the orientations of the lines induce a signature on the collection of all cells: to each cell we assign a sign-vector σ ∈ {−, 0, +} (We use “+” and “−” as shorthand for +1 and −1). The i-th entry σi of σ indicates whether the corresponding cell is on the positive side of li (σi = +), on the negative side of li (σi = −), or if the cell is entirely contained in li (σi = 0). For an