Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design

Given an undirected graph <i>G</i> = (<i>V,E</i>) withnonnegative costs on its edges, a root node <i>r</i> <i>V</i>, aset of demands <i>D</i> <i>V</i> with demand <i>v</i> <i>D</i>wishing to route <i>w(v)</i> units of flow (weight) to <i>r</i>,and a positive number <i>k</i>, the <i>Capacitated Minimum SteinerTree</i> (CMStT) problem asks for a minimum Steiner tree, rooted at<i>r</i>, spanning the vertices in <i>D</i> *&lcub;<i>r</i>&rcub;, in which the sum of the vertexweights in every subtree connected to <i>r</i> is at most <i>k</i>.When <i>D</i> &equals; <i>V</i>, this problem is known as the<i>Capacitated Minimum Spanning Tree</i> (CMST) problem. Both CMsTand CMST problems are NP-hard. In this article, we presentapproximation algorithms for these problems and several of theirvariants in network design. Our main results are the following: ---We present a (³ Á<sub><i>ST</i></sub> +2)-approximation algorithm for the CMStT problem, where ³ isthe <i>inverse Steiner ratio</i>, and Á<sub><i>ST</i></sub>is the best achievable approximation ratio for the Steiner treeproblem. Our ratio improves the current best ratio of2Á<sub><i>ST</i></sub> + 2 for this problem. ---In particular, we obtain (³ + 2)-approximation ratio forthe CMST problem, which is an improvement over the current bestratio of 4 for this problem. For points in Euclidean andrectilinear planes, our result translates into ratios of 3.1548 and3.5, respectively. ---For instances in the plane, under the<i>L</i><sub><i>p</i></sub> norm, with the vertices in <i>D</i>having uniform weights, we present a nontrivial(7/5Á<sub><i>ST</i></sub> + 3/2)-approximation algorithm forthe CMStT problem. This translates into a ratio of 2.9 for the CMSTproblem with uniform vertex weights in the<i>L</i><sub><i>p</i></sub>metric plane. Our ratio of 2.9 solvesthe long-standing open problem of obtaining any ratio better than 3for this case. ---For the CMST problem, we show how to obtain a 2-approximationfor graphs in metric spaces with unit vertex weights and <i>k</i> =3,4. ---For the <i>budgeted</i> CMST problem, in which the weights ofthe subtrees connected to <i>r</i> could be up to ± <i>k</i>instead of <i>k</i> (± e 1), we obtain a ratio of ³&plus; 2/±.

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