Minimum Partition of a Matroid Into Independent Subsets!
نویسنده
چکیده
A matroid M is a finite se t M of e le me nts with a famil y of subsets, called independent, such th a t (I) every subset of an independe nt se t is independent, and (2) for e ve ry subset A of M , all maximal indepe nde nt s ubsets of A have the sa me cardinality , called the rank r\A) of A. It is proved that a matroid can be partitioned into as few as k sets , each inde pende nt , if and only if every s ubse t A has cardinality at mos t k . r(A ).
منابع مشابه
The geometric lattice of embedded subsets
This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a partition where the former is a block of the latter. The lattice structure proposed in a recent contribution relies on ad-hoc definitions of both the join operator...
متن کاملAlgorithms and Data Structures for an Expanded Family of Matroid Intersection Problems
Consider a matroid of rank. n in which each element has a real-valued cost and one of d > I colors. A class of matroid intersection problems is studied in which one of the matroids is a partition matroid that specifies that a base have qj elements of color j. for j = I, 2•...• d. Relationships are characterized among the solutions to the family of problems generated when the vector (q l' q2' .....
متن کاملAn Algorithm for Determining Whether a given Binary Matroid Is Graphic
1. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following. A binary chain-group N on a. finite set M is a class of subsets of M forming a group under mod 2 addition. These subsets are the chains of N. A chain of N is elementary if it is non-null and has no other non-null chain of AT as a subset. A binary matroid is the class of ele...
متن کاملReducing the rank of a matroid
We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the minimum k-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at l...
متن کاملRandom sampling and greedy sparsification for matroid optimization problems
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...
متن کامل