Lecture Notes Discrete Optimization - Applied Mathematics

3. Matroids
3.1 Introduction
In the previous section, we have seen that the greedy algorithm can be used to solve the
MST problem. An immediate question that comes to ones mind is which other problems
can be solved by such an algorithm. In this section, we will see that the greedy algorithm
applies to a much broader class of optimization problems.
We first define the notion of an independent set system.
Definition 3.1. Let S be a finite set and let I be a collection of subsets of S. (S,I) is an
independent set system if
(M1) /0∈I ;
(M2) if I ∈I and J ⊆ I, then J ∈I.
Each set I ∈I is called an independent set; every other subset I ⊆ S with I /∈I is called a
dependent set. Further, suppose we are given a weight function w : S→R on the elements
in S.
Maximum Weight Independent Set Problem (MWIS):
Given:
Goal:
An independent set system(S,I) and a weight function w : S→R.
Find an independent set I ∈I of maximum weight w(I)=∑ x∈I w(x).
If w(x)