A Node Splitting Technique for Two Level Network Design Problems with Transition Nodes

Abstract

The Two Level Network Design (TLND) problem arises when local broadband access networks are planned in areas, where no existing infrastructure can be used, i.e., in the so-called greenfield deployments. Mixed strategies of Fiber-To-The-Home and Fiber-To-The-Curb, i.e., some customers are served by copper cables, some by fiber optic lines, can be modeled by an extension of the TLND. We are given two types of customers (primary and secondary), an additional set of Steiner nodes and fixed costs for installing either a primary or a secondary technology on each edge. The TLND problem seeks a minimum cost connected subgraph obeying a tree-tree topology, i.e., the primary nodes are connected by a rooted primary tree; the secondary nodes can be connected using both primary and secondary technology. In this paper we study an important extension of TLND in which additional transition costs need to be paid for intermediate facilities placed at the transition nodes, i.e., nodes where the change of technology takes place. The introduction of transition node costs leads to a problem with a rich structure permitting us to put in evidence reformulation techniques such as modeling in higher dimensional graphs (which in this case are based on a node splitting technique). We first provide a compact way of modeling intermediate facilities. We then present several generalizations of the facility-based inequalities involving an exponential number of constraints. Finally we show how to model the problem in an extended graph based on node splitting. Our main result states that the connectivity constraints on the splitted graph, projected back into the space of the variables of the original model, provide a new family of inequalities that implies, and even strictly dominates, all previously described cuts. We also provide a polynomial time separation algorithm for the more general cuts by calculating maximum flows on the splitted graph. We compare the proposed models both theoretically and computationally.