A clutter or antichain on a set defines a hypergraph. Matroid ports are a special class of clutters, and this paper deals with the diameter of matroid ports, that is, the diameter of the corresponding hypergraphs. Specifically, we prove that the diameter of every matroid port is at most 2. The main interest of our result is its application to secret sharing. Brickell and Davenport proved in 198...

For a matroid with an ordered (or “labelled”) basis, a basis exchange step removes one element with label l and replaces it by a new element that results in a new basis, and with the new element assigned label l. We prove that one labelled basis can be reconfigured to another if and only if for every label, the initial and final elements with that label lie in the same connected component of th...

A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes us...

Consider a matrix with m rows and n pairs of columns. The linear matroid parity problem (LMPP) is to determine a maximum number of pairs of columns that are linearly independent. We show how to solve the linear matroid parity problem as a sequence of matroid intersection problems. The algorithm runs in O(mn). Our algorithm is comparable to the best running time for the LMPP, and is far simpler ...

Over a finite field Fqm , the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew polynomials to the familiar concept of linear independence for vector spaces. This relation allows for the definition of a representable matroid called the Fqm[x;σ]-matroid, wi...

In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected solution weight is within a constant factor of th...

Graphic matroids form one of the most significant classes in matroid theory. When introducing matroids, Whitney concentrated on relations to graphs. The definition of some basic operations like deletion, contraction and direct sum were straightforward generalizations of the respective concepts in graph theory. Most matroid classes, for example those of binary, regular or graphic matroids, are c...

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in Σ 2 . In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be Σ 2 -complete and is coNPhard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism and matroid isomorphism a...

The inseparability graph of an oriented matroid is an invariant of its class of orientations. When an orientable matroid has exactly one class of orientations the inseparability graph of all its orientations is in fact determined by its non-oriented underlying matroid. From this point of view it is natural to ask if inseparability graphs can be used to characterize matroids which have exactly o...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization o...