Given a matroid equipped with a utility vector to define a matroid optimization problem, the corresponding inverse problem is to modify the utility vector as little as possible so that a given base of the matroid becomes optimal to the matroid optimization problem. The modifications can be measured by various distances. In this article, we consider the inverse matroid problem under the bottlene...

Let M = (E,F) be a matroid on a set E, and B one of its bases. A closed set θ ⊆ E is saturated with respect to B when |θ ∩B| = r(θ), where r(θ) is the rank of θ. The collection of subsets I of E such that |I ∩ θ| ≤ r(θ) for every closed saturated set θ turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by MB . In this paper we prove that a grap...

For any nite point set S in Ed, an oriented matroid DOM(S) can be de ned in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that the same construction of a Delaunay oriented matroid can be performed with respect to any smooth, stric...

The study of polyhedra within the framework of oriented matroids has become a natural approach. Methods for enumerating combinatorial types of convex polytopes inductively within the Euclidean setting alone have not been established. In contrast, the oriented matroid concept allows one to generate matroid polytopes inductively. Matroid polytopes, when not interesting in their own right as topol...

If ∆ is a polytope in real affine space, each edge of ∆ determines a reflection in the perpendicular bisector of the edge. The exchange group W (∆) is the group generated by these reflections, and ∆ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The GelfandSerganova Theorem and the st...

Properties of Boolean functions on the hypercube that are invariant with respect to linear transformations of the domain are among some of the most well-studied properties in the context of property testing. In this paper, we study the fundamental class of linear-invariant properties called matroid freeness properties. These properties have been conjectured to essentially coincide with all test...

We show that the infinite matroid intersection conjecture of NashWilliams implies the infinite Menger theorem proved recently by Aharoni and Berger. We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids. In particular, this proves the inf...

We prove that a binary matroid with huge branch-width contains the cycle matroid of a large grid as a minor. This implies that an infinite antichain of binary matroids cannot contain the cycle matroid of a planar graph. The result also holds for any other finite field. © 2007 Elsevier Inc. All rights reserved.

Let F4 be the root system associated with the 24-cell, and let M(F4) be the simple linear dependence matroid corresponding to this root system. We determine the automorphism group of this matroid and compare it to the Coxeter group W for the root system. We find non-geometric automorphisms that preserve the matroid but not the root system.

The most important open conjecture in the context of the matroid secretary problem claims the existence of an O(1)-competitive algorithm applicable to any matroid. Whereas this conjecture remains open, modified forms of it have been shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly at random [23, 20]. However, so far, no variant o...