Lecture Notes Discrete Optimization - Applied Mathematics

the MWIS problem is equivalent to finding a maximum weight basis of M.
We give some examples of matroids.
Example 3.2 (Uniform matroid). One of the simplest examples of a matroid is the socalled
uniform matroid. Suppose we are given some set S and an integer k. Define the
independent setsI as the set of all subsets of S of size at most k, i.e.,I ={I⊆ S | |I|≤k}.
It is easy to verify that M =(S,I) is a matroid. M is also called the k-uniform matroid.
Example 3.3 (Partition matroid). Another simple example of a matroid is the partition
matroid. Suppose S is partitioned into m sets S 1 ,...,S m and we are given m integers
k 1 ,...,k m . Define I = {I ⊆ S | |I∩ S i |≤k i for all 1≤i≤m}. Conditions (M1) and
(M2) are trivially satisfied. To see that Condition (M3) is satisfied as well, note that if
I,J ∈I and|I||I∩ S i | and thus
adding any element x ∈ S i ∩(J\ I) to I maintains independence. Thus, M = (S,I) is a
matroid.
Example 3.4 (Graphic matroid). Suppose we are given an undirected graph G=(V,E).
Let S=E and define I ={F ⊆ E | F induces a forest in G}. We already argued above
that Conditions (M1) and (M2) are satisfied. We next show that Conditions (M4) is satisfied
too. Let U ⊆ E. Consider the subgraph(V,U) of G induced by U and suppose that it
consists of k components. By definition, each basis B of U is an inclusionwise maximal
forest contained in U. Thus, B consists of k spanning trees, one for each component of
the subgraph (V,U). We conclude that B contains |V|−k elements. Because this holds
for every basis of U, Condition (M4) is satisfied. We remark that any matroid M =(S,I)
obtained in this way is also called a graphic matroid (or cycle matroid).
Example 3.5 (Matching matroid). The independent set system of Example 3.1 is not a
matroid. However, there is another way of defining a matroid based on matchings. Let
G = (V,E) be an undirected graph. Given a matching M ⊆ E of G, let V(M) refer to
the set of nodes that are incident to the edges of M. A node set I ⊆ V is covered by
M if I ⊆ V(M). Define S = V and I = {I ⊆ V | I is covered by some matching M}.
Condition (M1) holds trivially. Condition (M2) is satisfied because if a node set I ∈I is
covered by a matching M, then each subset J ⊆ I is also covered by M and thus J ∈I. It
can also be shown that Condition (M3) is satisfied and thus M =(S,I) is a matroid. M is
also called a matching matroid.
3.3 Greedy Algorithm for Matroids
The next theorem shows that the greedy algorithm given in Algorithm 3 always computes
a maximum weight independent set if the underlying independent set system is a matroid.
The theorem actually shows something much stronger: Matroids are precisely the
independent set systems for which the greedy algorithm computes an optimal solution.
Theorem 3.1. Let (S,I) be an independent set system. Further, let w : S → R + be a
nonnegative weight function on S. The greedy algorithm (Algorithm 3) computes an independent
set of maximum weight if and only if M =(S,I) is a matroid.
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