Interior Point Algorithms for Network Ow Problems

A large number of problems in transportation, communications and manufacturing can be modelled as network ow problems. In these problems one seeks to nd the most e cient, or optimal, way to move ow (e.g. materials, information, buses, electrical currents) on a network (e.g. postal network, computer network, transportation grid, power grid). Many of these optimization problems are special classes of linear programming problems, with combinatorial properties that allow the development of e cient solution techniques. In this chapter, we limit our discussion to linear network ow problems. For a treatment of non-linear network ow problems, the reader is referred to [17, 28, 29, 48]. Given a directed graph G = (N ;A), where N is a set of m nodes and A a set of n arcs, let (i; j) denote a directed arc from node i to node j. Every node is classi ed in one of the following three categories. Source nodes produce more ow than they consume. Sink nodes consume more ow than they produce. Transshipment nodes produce as much ow as they consume. Without loss of generality, one can assume that the total ow produced in the network equals the total ow consumed. Each arc has associated with it an origin node and a destination node, implying a direction for ow to follow. Arcs have limitations (often called capacities or bounds) on how much ow can move through them. The ow on arc (i; j) must be no less than lij and can be no greater than uij. To set up the problem in the framework of an optimization problem, a unit ow cost cij , incurred by each unit of ow moving through arc (i; j), must be de ned. Besides being restricted by lower and upper bounds at each arc, ows must satisfy another important condition, known as Kirchho 's Law (conservation of ow), which states that for every node in the network, the sum of all incoming ow plus the ow produced at the node must equal the sum of all outgoing ow plus the ow consumed at the node. The objective of the minimum cost network ow problem is to determine the ow on each arc of the network, such that all