In which we look at the linear programming formulation of the maximum flow problem, construct its dual, and find a randomized-rounding proof of the max flow – min cut theorem.
In the first part of the course, we designed approximation algorithms “by hand,” following our combinatorial intuition about the problems. Then we looked at linear programming relaxations of the problems we worked on, and we saw that approximation algorithms for those problems could also be derived by rounding a linear programming solution. We also saw that our algorithms could be interpreted as constructing, at the same time, an integral primal solution and a feasible solution for the dual problem.
Now that we have developed exact combinatorial algorithms for a few problems (maximum flow, minimum s-t cut, global min cut, maximum matching and minimum vertex cover in bipartite graphs), we are going to look at linear programming relaxations of those problems, and use them to gain a deeper understanding of the problems and of our algorithms.
We start with the maximum flow and the minimum cut problems.