4. Lecture Notes on Matroid Optimization 4.1 Definition of a Matroid
نویسنده
چکیده
Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is denoted by I or I(M), and we typically refer to a matroid M by listing its ground set and its family of independent sets: M = (E, I). For M to be a matroid, I must satisfy two main axioms: (I1) if X ⊆ Y and Y ∈ I then X ∈ I, (I2) if X ∈ I and Y ∈ I and |Y | > |X| then ∃e ∈ Y \X : X ∪ {e} ∈ I.
منابع مشابه
Michel X . Goemans 4 . Lecture notes on matroid optimization 4 . 1 Definition of a Matroid
Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is de...
متن کاملInstructor : Chandra Chekuri Scribe : Sreeram Kannan 1 Maximum Weight Independent Set in a Matroid , Greedy Algo - rithm , Independence and Base Polytopes
We saw the definition of base, circuit, rank, span and flat of matroids last lecture. We begin this lecture by studying some more basic properties of a matroid. Exercise 1 Show that a set I ⊆ S is independent in a matroid M iff ∀y ∈ I, there exists a flat F such that I − y ⊆ F and y ∈ F. Exercise 3 Verify that (S, I *) is indeed a matroid. Remark 4 Having an independence or rank oracle for M im...
متن کامل5. Lecture Notes on Matroid Intersection
One nice feature about matroids is that a simple greedy algorithm allows to optimize over its independent sets or over its bases. At the same time, this shows the limitation of the use of matroids: for many combinatorial optimization problems, the greedy algorithm does not provide an optimum solution. Yet, as we will show in this chapter, the expressive power of matroids become much greater onc...
متن کاملMatroid Representation , Geometry and Matrices
The connections between algebra and finite geometry are very old, with theorems about configurations of points dating to ancient Greece. In these notes, we will put a matroid theoretic spin on these results, with matroid representations playing the central role. Recall the definition of a matroid via independent sets I. Definition 1.1. Let E be a finite set and let I be a family of subsets of E...
متن کامل