# Matchings, Cuts, and Flows

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- Busacker, R. G., Coffin, S. A., and Gowen, P. J. (1962). Three general network flow problems and their solutions. Staff Paper RAC-SP-183, Research Analysis Corporation, Operations Logistics Division. [http://handle.dtic.mil/100.2/AD296365]
- Ford, L. R. and Fulkerson, D. R. (1957). A simple algorithm for finding maximal network flows and an application to the hitchcock problem. Canadian Journal of Mathematics, 9:210–218. [http://smc.math.ca/cjm/v9/p210]MathSciNetMATHCrossRefGoogle Scholar
- Ford, L. R. and Fulkerson, D. R. (1962).
*Flows in networks*. Technical Report R-375-PR, RAND Corporation. [http://www.rand.org/pubs/reports/R375] - Jungnickel, D. (2007).
*Graphs, Networks and Algorithms*. Springer, third edition.Google Scholar - Goh, C. J. and Yang, X. Q. (2002).
*Duality in Optimization and Variational Inequalities*. Optimization Theory and Applications. Taylor & Francis.Google Scholar - Goldberg, A. V. and Kennedy, R. (1995). An efficient co st scaling algorithm for the assignment problem.
*Mathematical Programming*, 71:153–178. [http://theory.stanford.edu/~robert/papers/csa.ps]MathSciNetMATHGoogle Scholar - Schwartz, B. L. (1966). Possible winners in partially completed tournaments.
*SIAM Review*, 8(3):302–308. [http://jstor.org/pss/2028206]MATHCrossRefGoogle Scholar