# Matchings, Cuts, and Flows

• Magnus Lie Hetland

## Abstract

While the previous chapter gave you several algorithms for a single problem, this chapter describes a single algorithm with many variations and applications. The core problem is that of finding maximum flow in a network, and the main solution strategy I’ll be using is the augmenting path method of Ford and Fulkerson. Before tackling the full problem, I’ll guide you through two simpler problems, which are basically special cases (they’re easily reduced to maximum flow). These problems, bipartite matching and disjoint paths, have many applications themselves and can be solved by more specialized algorithms. You’ll also see that the max-flow problem has a dual, the min-cut problem, which means that you’ll automatically solve both problems at the same time. The min-cut problem has several interesting applications that seem very different from those of max-flow, even if they are really closely related. Finally, I’ll give you some pointers on one way of extending the max-flow problem, by adding costs, and looking for the cheapest of the maximum flows, paving the way for applications such as mincost bipartite matching.

## Keywords

Shipping Rounded Metaphor Vanilla
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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