1. Network Flow

2. 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. 2301233TREES2 st uv xy 2 5 16 2 5 4 4 3

3. Network Flows Flow f : E  R + such that Value of flow f is 2301233TREES3

4. Example 2301233TREES4 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

5. Max Flow Problem Given G, s and t, determine max-valued flow from s to t. 2301233TREES5 st uv xy 2/2 4/5 0/1 2/6 2/2 0/5 4/4 2/3

6. Greedy Method 2301233TREES6 s t uv xy 25 16 2 5 4 4 3 3 2 2

7. Cut A cut (S, T) of a flow network G =(V, E) is a partition of V such that s  S and t  T. 2301233TREES7 st uv xy 2 5 1 6 2 5 4 4 3

8. 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. 2301233TREES8 st uv xy 2 5 1 6 2 5 4 4 3 6 8 106 7 8 8

9. Min Cut Problem Given a network G with capacity c, and vertices s and t, determine the minimum- capacity cut. 2301233TREES9 st uv xy 2 5 1 6 2 5 4 4 3 6 8 106 7 8 8

10. Max flow/ Min cut For any network G with capacity c, the value of the maximal flow is equal to the minimum- capacity cut. 2301233TREES10 st uv xy 2/2 4/5 0/1 0/6 2/2 0/5 4/4 2/3 6