In this lecture, the focus is on submodular function in combinatorial optimizations. The first class of submodular functions which was studied thoroughly was the class of matroid rank functions. The flourishing stage of matroid theory came with Jack Edmonds’ work in 1960s, when he gave a minmax formula and an efficient algorithm to the matroid partition problem, from which the matroid intersect...

A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matr...

In Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are B-shellable. Our result shows that the whole Oriented Matroid is shellable, too.

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...