Network Flows
When I do these lectures live, it normally takes me 3 lectures to go through the material slowly and do lots of examples. This is a list of topics that are needed to under network flows. You goal should be to understand each topic.
- Definition of flow network.
- The minimum-cut problem.
- The maximum flow problem.
- The definition of a residual network.
- An Augmented path.
- The statement of the Min-cut, max-flow theorem.
- The Ford–Fulkerson algorithm.
- How to use the max-flow algorithm for maximum bipartite matching.
Lecture 8
- Introduction to NetworkFlow I Slides 1-23 and Demo of Ford-Fulkerson
Lecture 9
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Alternative mathematical API. Flows that are Skew Symmetric flows, that is $f(u,v)= - f(v,u)$. We’ll do this on the blackboard, but skew symmetric flows are a bit counter intuitive, but they make the proofs and implementation easier. Take a look at the presentation at Wikipedia. This is not examined, and I won’t go into much detail.
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Min-Cut Max-Flow duality slides 25-36 of NetworkFlow I
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Bipartite matching slides 1-17 of NetworkFlow II.
Reading Guide
What should I know by the end of this set of lectures?
- Do I understand the definitions of a flow network, min-cuts and maximum flows?
- Why does the naive greedy algorithm for maximum flow fail?
- The relation between augmented paths and residual networks.
- What is the relationship between flows and cuts?
- What is the relationship between negative flows and residual networks?
- What does the min-cut max-flow algorithm tell us and how does it help us to derive an algorithm?
- What is bipartite matching? What are its applications?