Flows in Networks

Abstract

The purpose of this chapter is to describe basic elements of the theory and applications of network flows. This topic is probably the most important single tool for applications of digraphs and perhaps even of graphs as a whole. At the same time, from a theoretical point of view, flow problems constitute a beautiful common generalization of shortest path problems and problems such as finding internally (arc)-disjoint paths from a given vertex to another. The theory of flows is well understood and fairly simple. This, combined with the enormous applicability to real-life problems, makes flows a very attractive topic to study. From a theoretical point of view, flows are well understood as far as the basic questions, such as finding a maximum flow from a given source to a given sink or characterizing the size of such a flow, are concerned. However, the topic is still a very active research field and there are challenging open problems such as deciding whether an O(nm) algorithm exists for the general maximum flow problem.