Circulations

In Chapter 6, we introduced the simplest kind of flow problems, namely the determination of maximal flows in a network; and in Chapter 7, we studied various applications of this theory. The present chapter deals with generalizations of the flows we worked with so far. For example, quite often there are also lower bounds on the capacities of the edges given, or a cost function on the edges. To solve this kind of problem, it makes sense to remove the exceptional role of the vertices s and t by requiring the flow preservation condition (F2) of Chapter 6 for all vertices, including s and t. This leads to the notion of circulations on directed graphs. As we will see, there are many interesting applications of this more general concept. To a large part, these cannot be treated using the theory of maximal flows as presented before; nevertheless, the methods of Chapter 6 will serve as fundamental tools for the more general setting.