Max-flow Min-cut Theorem

Given a bipartite graph G and a matching M, the following are equivalent:

(1)

The current matching is maximum.

(2)

There are no augmenting paths.

(3)

The size of the matching is equal to the size of some cut.

Proof:

• (1) implies (2) because the existence of an augmenting path implies that the current matching is not maximum. So maximum implies no augmenting path.
• (3) implies (1) because of the lemma: the size of a cut upper bounds the size of the matching, so if we have a matching that meets this upper bound, it is maximum.
• (2) implies (3) is the interesting one...