Many problems that can be correctly solved by greedy algorithms can be described in terms of an abstract combinatorial object called a matroid. Matroids were first described in 1935 by the mathematician Hassler Whitney as a combinatorial generalization of linear independence of vectors—‘matroid’ means ‘something sort of like a matrix’. A matroid M is a finite collection of finite sets that sati...

We prove that, if M is a weakly 4-connected matroid with |E(M)| 7 and neither M nor M∗ is isomorphic to the cycle matroid of a ladder, then M has a proper minor M ′ such that M ′ is weakly 4-connected and |E(M ′)| |E(M)| − 2 unless M is some 12-element matroid with a special structure. © 2007 Elsevier Inc. All rights reserved.

The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown NP -hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. Also the corresponding approximation problem is shown NP -hard for certain approximation performance bounds. This is in contrast with the maximum matroid-matroid intersection ...

Let E be a possibly infinite set and let M N matroids defined on E. We say that the pair {M,N} has Intersection property if share an independent I admitting bipartition IM⊔IN such spanM(IM)∪spanN(IN)=E. The Matroid Conjecture of Nash-Williams says every matroid property. conjecture is known easy to prove in case when one uniform it was shown by Bowler Carmesin implied its special where direct s...

For each proper minor-closed subclassM of the GF(q)representable matroids containing all GF(q)-representable matroids, we give, for all large r, a tight upper bound on the number of points in a rank-r matroid inM, and give a rank-r matroid inM for which equality holds. As a consequence, we give a tight upper bound on the number of points in a GF(q)-representable, rank-r matroid of large rank wi...

A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible hvectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize...

Las Vergnas Cube Conjecture states that the cube matroid has exactly one class of orientations. We prove that this conjecture is equivalent to saying that the oriented matroid Aff(C), of the affine dependencies of the n-cube C := {−1, 1} over IR, can be reconstructed from the underlying matroid and one of the following partial lists of signed circuits or cocircuits: 1) the signed circuits of ra...

Consider the following online version of the submodular maximization problem under a matroid constraint: We are given a set of elements over which a matroid is defined. The goal is to incrementally choose a subset that remains independent in the matroid over time. At each time, a new weighted rank function of a different matroid (one per time) over the same elements is presented; the algorithm ...