Monday, 2 November 2015

Lecture 10

I have been working on the notes for this lecture and have now added several further figures. Please make sure you have an up-to-date copy of the notes, as there were some errors in previous versions.

I will talk about the Hungarian algorithm for the assignment problem at the start of next lecture.

I mentioned that in a problem where all edge capacities are integers the Ford-Fulkerson algorithm runs in time $O(|E|\cdot f)$, where $f$ is some bound on the total flow. If edge capacities are not integers, one can still apply the Ford-Fulkerson algorithm, but it is not guaranteed to converge. There are examples in which one can increase flows on augmenting paths by ever smaller and smaller amounts, but not even converge to the optimal flow.

To understand the proofs of Hall's marriage theorem and Konig's theorem, you really need to sit down and carefully study for yourself an example, as I have now provided in Figure 12.

There are many nice applications in which the min-cut max-flow theorem can be used to obtain other results, by applying it to the right network. Another example of this is Menger's theorem: Let $G$ be a finite undirected graph and $x$ and $y$ two distinct vertices. Then the minimum number of edges whose removal disconnects $x$ and $y$ is equal to the maximum number of pairwise edge-independent paths from $x$ to $y$. This theorem is proved similarly to Konig's theorem.