A constant-ratio approximation algorithm for a class of hub-and-spoke network design problems and metric labeling problems: star metric case

Transportation networks frequently employ hub-and-spoke network architectures to route flows between many origin and destination pairs. Hub facilities work as switching points for flows in large networks. In this study, we deal with a problem, called the single allocation hub-and-spoke network design problem. In the problem, the goal is to allocate each non-hub node to exactly one of given hub nodes so as to minimize the total transportation cost. The problem is essentially equivalent to another combinatorial optimization problem, called the metric labeling problem. The metric labeling problem was first introduced by Kleinberg and Tardos in 2002, motivated by application to segmentation problems in computer vision and related areas. In this study, we deal with the case where the set of hubs forms a star, which arises especially in telecommunication networks. We propose a polynomial-time randomized approximation algorithm for the problem, whose approximation ratio is less than 5.281. Our algorithms solve a linear relaxation problem and apply dependent rounding procedures.