LP extreme points and cuts for the fixed-charge network design problem

Many design decisions in transporation, communication, and manufacturing planning can be modeled as problems of routing multiple commodities between various origin and destination nodes of a directed network. Each arc of the network is uncapacitated and carries a fixed charge as well as a cost per unit of flow. We refer to the general problem of selecting a subset of arcs and routing the required multi-commodity flows along the chosen arcs at a minimum total cost as the fixed charge network design problem. This paper focuses on strenghthening the linear programming relaxation of a path-flow formulation for this problem. The considerable success achieved by researchers in solving many related design problems with algorithms that use strong linear programming-based lower bounds motivates this study. We first develop a convenient characterization of fractional extreme points for the network design linear programming relaxation. An auxiliary graph introduced for this characterization also serves to generate two families of cuts that exclude some fractional solutions without eliminating any feasible integer solutions. We discuss a separation procedure for one class of inequalities and demonstrate that many of our results generalize known properties of the plant location problem. Supported in part by grant number ECS-831-6224 of the National Science Foundation.