Approximation algorithms for the single allocation problem in hub-and-spoke networks and related metric labeling problems

This paper deals with a single allocation problem in hub-and-spoke networks. We present a simple deterministic 3-approximation algorithm and randomized 2-approximation algorithm based on a linear relaxation problem and a randomized rounding procedure. We handle the case where the number of hubs is three, which is known to be NP-hard, and present a (5/4)-approximation algorithm. The single allocation problem includes a special class of the metric labeling problem, defined by introducing an assumption that both objects and labels are embedded in a common metric space. Under this assumption, we can apply our algorithms to the metric labeling problem without loosing theoretical approximation ratios. As a byproduct, we also obtain a (4/3)-approximation algorithm for an ordinary metric labeling problem with three labels.