Max-Flow Min-Cut: Reconciling Graph Theory with Linear Programming

Network ?ow theory constitutes a thoroughly studied mathematical ?eld which is widely applied in practice in areas as diverse as economics, traffic routing, or telecommunication. Interestingly, networks can be discussed equally well by using methods from graph theory or from linear programming (LP). Both of these approaches have advantages and disadvantages, and are usually treated using their own terminology and techniques. In this piece of work, the notions of network theory are developed such that the combinatorial and the LP point of view can be presented as homogeneously as possible. All de?nitions and theorems are formulated using a fairly general network model that lends itself well to both disciplines. After the introduction of the basic ideas, the central theorem of network flow theory, the Max-Flow Min-Cut Theorem, is revised. Then some interesting existence results and algorithms for flow maximization are looked at. A section with applications concludes the text. All defininitions from graph theory and LP that are needed are included within the book, which is hence easily accessible to everyone with a basic knowledge of algebra and analysis.