# Lecture 23

You may want to download the lecture slides that were used for these videos (PDF).

## 1. Network Flow Problems

This video states a network flow problem that we will explore in this lecture. (4:05)

## 2. A Flow in a Network

This video defines a flow. (4:28)

## 3. Challenge: Find a Maximum Flow (1)

Finding a flow is trivial. Finding a good flow might be very hard. (3:29)

## 4. Challenge: Find a Maximum Flow (2)

Can you look at the graph in this video, and identify a flow that is higher than 99? (5:11)

## 5. Challenge: Find a Maximum Flow (3)

This video introduces the hatcheck problem. (3:31)

## 6. An Augmenting Path

Suppose we have a network, and we can find a path that does from the source to the sink. Every edge on that path has a flow and a capacity. What does it mean if an edge is full, or if it is empty? If all of the edges are not full, can we change the flow in a way to increase the value? (6:40)

## 7. Generalizing our Notion of an Augmenting Path

In this video we strengthen and generalize our notion of an augmenting path. (6:34)

## 8. Can we Find an Augmenting Path?

With our more general notion of our augmenting path, we return to our network, and look for an augmenting path. (4:50)

## 9. Can we Find an Upper Bound on the Flow?

Now we’re at 107, and we want to get the flow up to 120. How can we reach this value? Do we need to lower our expectations? (1:39)

## 10. A Cut

This video introduces the concept of a cut in a network flow. (3:08)

## 11. Cuts and Capacity

We cannot have a flow that exceeds the capacity of a cut. (4:31)

## 12. Can You Identify a Cut Whose Capacity is 107?

See if you can now identify a cut whose capacity is 107. (2:38)

At the beginning of the following lecture, Dr. Trotter reviewed some of this lecture’s material. You may find the following videos helpful in getting a better understanding of the material in Lecture 23.

## 13. Review – A Network Flow Problem

This video reviews the network flow problem, and what our goals are. (2:07)

## 14. Review – Cuts in Network Flow Problems

This video reviews cuts, which separate the verticies into two parts. (1:45)

## 15. Review – The Max Flow/Min Cut Theorem

The maximum value of a flow is equal to the minimum capacity of a cut. (1:29)

## 16. Review – Full and Empty Edges

An edge is full when the flow on the edge is equal to the capacity of the edge. An edge is empty when the flow on the edge is 0. (1:59)

## 17. Review – Augmenting Paths

Allowing the ability to walk on an edge in the network in either direction, an ordinary path from S to T traverses some edges in a forward manner and others in a backwards manner. The first and last are always forward. The path is called an augmenting path when the forward edges are not full and the backwards edges are not empty. (5:20)