Lecture 6

Notes below are meant to supplement the scribed notes.

Max-flow Min-cut Problem (Section 6.4)

The max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e. the smallest total weight of the edges which if removed would disconnect the source from the sink.

To find the minimum cut in a graph, it suffices to run max-flow algorithms with a fixed source and all possible sinks (i.e. run max-flow for \(|V|-1\) times), and then take the minimum over all computed cuts.

Correctness: Let the true global minimum cut be \((S^*,T^*)\) such that \(S^*\cup T^* = V\) and \(S^*\cap T^* = \emptyset\). Suppose the fixed source is \(s_0\), and assume w.l.o.g. that \(s_0\in S^*\). Since \(T^*\) is non-empty, there exists at least one vertex \(t_0\in T^*\). When we run the max-flow algorithm with source \(s_0\) and sink \(t_0\), the min-cut found by the algorithm must be equal to the true global minimum cut \((S^*,T^*)\).

Count edges of subgraphs (Corollary 6.3): For an \(n\)-vertex graph with minimum cut of size \(k\), the number of edges is at least \(\frac{kn}{2}\). This is because each vertex has degree at least \(k\) (to avoid a cut of size less than \(k\)).