Augmenting Path Theorem: A flow f is maximum iff it admits no augmenting path
Already saw that if flow admits an augmenting path, then it is not maximum.
Suppose f admits no augmenting path. Need to show f maximum.
Cut -- a set of edges that separate s from t.
Capacity of a cut = sum of capacities of edges in cut.
Prove theorem by showing that there is a cut whose capacity = f.
A = vertices s.t. for each v in A, there is augmenting path from s to v. => defines a cut.
Claim: for all edges in cut, f(v,w)=c(v,w)
=> value of flow = capacity of cut defined by A => maximum.