Difficulty in understand the proof of the lemma : “Matroids exhibit the optimal-substructure property”


I was going through the text "Introduction to Algorithms" by Cormen et. al. where I came across a lemma in which I could not understand a vital step in the proof. Before going into the lemma I give a brief description of the possible prerequisites for the lemma.


Let $ M=(S,\ell)$ be a matroid where $ S$ is the ground set and $ \ell$ is the family of subsets of $ S$ called the independent subsets of $ S$ .

Let us have an algorithm which finds an optimal subset of $ M$ using greedy method as:

$ GREEDY(M,W):$

$ 1\quad A\leftarrow\phi$

$ 2\quad \text{sort $ S[M]$ into monotonically decreasing order by weight $ w$ }$

$ 3\quad \text{for each $ x\in S[M]$ , taken in monotonically decreasing order by weight $ w(x)$ }$

$ 4\quad\quad \text{do if $ A\cup\{x\} \in \ell[M]$ }$

$ 5\quad\quad\quad\text{then $ A\leftarrow A\cup \{x\}$ }$

$ 6\quad \text{return $ A$ }$


I was having a problem in understanding a step in the proof of the lemma below.

Lemma: (Matroids exhibit the optimal-substructure property)

Let $ x$ be the first element of $ S$ chosen by $ GREEDY$ for the weighted matroid $ M = (S, \ell)$ . The remaining problem of finding a maximum-weight independent subset containing $ x$ reduces to finding a maximum-weight independent subset of the weighted matroid $ M’ = (S’, \ell’)$ , where

$ S’ = \{y\in S:\{x,y\}\in \ell\}$ ,

$ \ell’ = \{В \subseteq S – \{x\} : В \cup \{x\} \in \ell\}$ ,

and the weight function for $ M’$ is the weight function for $ M$ , restricted to $ S’$ . (We call $ M’$ the contraction of $ M$ by the element $ x$ .)

Proof:

  1. If $ A$ is any maximum-weight independent subset of $ M$ containing $ x$ , then $ A’ = A — \{x\}$ is an independent subset of $ M’$ .

  2. Conversely, any independent subsubset $ A’$ of $ M’$ yields an independent subset $ A = A’\cup\{x\}$ of $ M$ .

  3. We have in both cases $ w(A) = w(A’) + w(x)$ .

  4. Since we have in both cases that $ w(A) = w(A’) + w(x)$ , a maximum-weight solution in $ M$ containing $ x$ yields a maximum-weight solution in $ M’$ , and vice versa.


I could understand $ (1),(2),(3)$ . But I could not get how the line $ (4)$ was arrived in the proof from $ (1),(2),(3)$ . especially the part in bold-italics. Could anyone please make it clear to me.