Lecture notes 7: Network flow problems

We now consider network flow problems. Such problems have some very nice properties. Specifically, we consider the minimum cost network flow problem, also known as the transshipment problem. In this problem, we are given a directed graph (V,E). The goal is to transport certain amounts of a single commodity from some vertices to others, across the edges of the graph. Specifically, each vertex v has a demand dv, which is the net amount of the commodity that it wants to receive. dv can be negative: in this case, the vertex has a net supply rather than a net demand, that is, the vertex is a producer of the commodity. We assume that demand and supply are perfectly matched overall, that is, ∑ v dv = 0. We can transport an unlimited amount across each edge, but associated with each edge e, there is a cost ce of transporting one unit across that edge. Our goal is to find a flow such that everyone’s demand is met (exactly), while incurring the minimum total transportation cost. We can without loss of generality assume that there are edges in both directions between any two vertices (if there is no such edge originally, we can add one with (near) infinite cost). We can write this as the following linear program: