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...

This paper presents a theorem concerning a matroid with the parity condition. The theorem provides matroid generalizations of graph-theoretic results of Lewin [3] and Gallai [1]. Let M=(E, F) be a matroid, where E is a finite set of elements and F is the family of independent sets of M (in this paper, we presuppose a knowledge of matroid theory; our standard reference is Welsh [4]). Assume that...

Let k be an integer exceeding one. The class of k–regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank–r regular matroid has the maximum number of points if and only if it is isomorphic to M(Kr+1), the cycle matroid of the complete graph on r + 1 vertices. A simple rank–r near-regular matroid has the maximum number of points if and only if it i...

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...

The closed fuzzy matroid is a subsystem of the fuzzy matroid theory. Some important conclusions, ideas and methods of the closed fuzzy matroid are briefly introduced and commented in this paper. We try to let readers to know the latest development and results of the closed fuzzy matroids.

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.