Network Flow

Network flow formulation A network G = (V, E). Capacity c(u, v)  0 for edge (u, v). Assume c(u, v) = 0 if (u, v)  E. Source s and sink t TREES2 st uv xy

Network Flows Flow f : E  R + such that Value of flow f is TREES3

Example TREES4 st uv xy 1/2 3/5 0/1 2/6 2/2 0/5 4/4 3/4 2/3 Capacity constraint Flow conservation

Max Flow Problem Given G, s and t, determine max-valued flow from s to t TREES5 st uv xy 2/2 4/5 0/1 2/6 2/2 0/5 4/4 2/3

Greedy Method TREES6 s t uv xy

Cut A cut (S, T) of a flow network G =(V, E) is a partition of V such that s  S and t  T TREES7 st uv xy

Capacity of a Cut The capacity of a cut (S, T) is the sum of the capacity of all edges (u, v) such that u  S and v  T TREES8 st uv xy

Min Cut Problem Given a network G with capacity c, and vertices s and t, determine the minimum- capacity cut TREES9 st uv xy

Max flow/ Min cut For any network G with capacity c, the value of the maximal flow is equal to the minimum- capacity cut TREES10 st uv xy 2/2 4/5 0/1 0/6 2/2 0/5 4/4 2/3 6