Man Muss Immer Generalisieren. [one Must Always Generalize.] — Attributed to Balances and Pseudoflows F.. Unbalanced Flows

In this chapter, we consider a generalization of flows that allows “stuff” to be injected or extracted from the flow network at the vertices. For the moment, consider a flow network G = (V, E) without any specific source and target vertices. Let b : V → R be a balance function describing how much flow should be injected (if the value is positive) or extracted (if the value is negative) at each vertex. We interpret positive balances as demand or deficit and negative balances as (negated) supply or excess. We now redefine the word flow to mean a function f : E → R that satisfies the modified balance condition