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

Authors:
Masaru Iwasa;Hiroo Saito;Tomomi Matsui
Affiliations:
Graduate School of Information Science and Technology, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan;Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan;Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

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 losing theoretical approximation ratios. As a byproduct, we also obtain a (4/3)-approximation algorithm for an ordinary metric labeling problem with three labels.