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. In this paper, we deal with a problem, called the single allocation hub-and-spoke network design problem. In the single allocation hub-andspoke network design 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 [29] in 2002, motivated by application to segmentation problems in computer vision and energy minimization problems in related areas. In this paper, we deal with the case where the set of hubs forms a star, which is called the star-star hub-and-spoke network design problem, and the star-metric labeling problem. This model arises especially in telecommunication networks in the case where set-up costs of hub links are considerably large or full interconnection is not required. We propose a polynomialtime randomized approximation algorithm for these problems, whose approximation ratio is less than 5.281. Our algorithms solve a linear relaxation problem and apply dependent rounding procedures.