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 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.