Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the eeectiveness of these paradigms for two optimization problems on matroids: nding an optimum matroid basis and packing disjoint matroid bases. Applications of th...

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.

We show that the set of r-quasi-transversals of a matroid, if nonempty, is the set of bases of a matroid. We also give an alternative proof of the known theorem which identifies the conjugate of the rank partition of a matroid.

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.

In this paper we provide two recognition algorithms for the class of signed-graphic matroids along with necessary and sufficient conditions for a matroid to be signed-graphic. Specifically, we provide a polynomial-time algorithm which determines whether a given binary matroid is signed-graphic and an algorithm which determines whether a general matroid given by an independece oracle is binary s...

Gioan introduced the circuit-cocircuit reversal system of an oriented matroid and showed that its cardinality equals the number of bases when the underlying matroid is regular. We prove that the equality fails whenever the underlying matroid is not regular, hence giving a new characterization of regular matroids.

In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems. The connectivity of a matroid is an important measure of its robustness as a network. Therefore, it is very necessary ...