Reformulation, Column Generation and Lagrangian Relaxation for Local Access Network Design Problems

In this paper, we present several models of Local Access Network Design and Expansion problems, and discuss different alternatives for solving these problems, as well as the relationships among different approaches. One of our results shows that both Design and Expansion problems provide examples for a rare situation in which the lower bound from Lagrangian relaxation is exactly equal to the optimal value of integer programming while the lower bound from its linear programming relaxation is inferior. We reformulate both Design and Expansion problems, via multicommodity flow reformulation, to an (implicitly defined) linear program which has Dantzig-Wolf block structure. Then, we illustrate that the so-called the Limited Column Generation procedure can be interpreted as a dedicated Dantzig-Wolf decomposition procedure. Therefore, a relationship among the Limited Column Generation, multicommodity flow reformulation, and Dantzig-Wolf Decomposition is established. Subject classification: Networks/graphs: applications, theory, tree algorithms. Programming: decomposition, column generation. Reformulation, Column Generations and Lagrangian Relaxation for Local Access Network Design Problems Dong Xiao Shaw This paper discusses different approaches for solving the Local Access Network Design and Expansion problems. A local access network connects customer nodes to a switching center (central office) which, in turn, communicates with other switching centers via the inter-office and backbone networks. Most existing local access networks have a tree structure, i.e., each customer node is connected via a unique path to the switching center. Effective design and expansion planning of local access networks can yield a significant savings in cost. Such problems have attracted much attentions in recent years. The discussion of the modeling issues can be found in Balakrishnan et al (1991), and a Lagrangian relaxation based approach is developed by Balakrishnan, Magnanti and Wong (1991). Aghezzaf, Magnanti and Wolsey (1992) studied the combinatorial structure of the model. A related, but different model was formulated by Cook (1990), and a solution procedure and related knapsack-type problem were studied by Bienstock (1993a and b). Recently, a so-called Limited Column Generation has been proposed by Shaw (1993) to solve the Local Access Network Design and Expansion problems. An efficient implementation of such an algorithm is developed in Cho and Shaw (1994b). The computational experiments in Cho and Shaw (1994b) indicate that such an approach is very efficient in practice, instances of the Expansion problem with 50 nodes can be solved in 175 CPU seconds on average and instances with 100 nodes can be solved in 1020 CPU seconds on average on a SUN SPARC 1000 workstation. The success of the Limited Column Generation approach motivated the research in this paper. Here, we discuss several computational aspects and the relationships among different approaches.