Max-flow Min-cut Theorem

Augmenting path theorem. Flow f is a max flow iff there are no augmenting paths. Max-flow min-cut theorem. [Ford-Fulkerson 1956] The value of the max flow is equal to the value of the min cut. Proof strategy. We prove both simultaneously by showing the TFAE: (i) There exists a cut (A, B) such that v(f) = cap(A, B). (ii) Flow f is a max flow. (iii) There is no augmenting path relative to f. (i) ⇒ (ii) This was the corollary to weak duality lemma. (ii) ⇒ (iii) We show contrapositive. ! Let f be a flow. If there exists an augmenting path, then we can improve f by sending flow along path.