Lecture : Supplies, Demands, and Pseudoflows [sp'''] .. Reduction to Maximum Flow

In this note, we consider a mild generalization of flows, which allow a fixed amount of flow to be injected or extracted from the flow network at the vertices. For the moment, consider a flow network G = (V, E) without identified 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 ∑