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

TL;DR: Approximation algorithms for these problems and several of theirvariants in network design are presented, including a nontrivial (7/5Á_<i>ST</i> + 3/2)-approximation algorithm for the CMST problem and the best achievable approximation ratio for the Steiner treeproblem.

Abstract: Given an undirected graph G = (V,E) with nonnegative costs on its edges, a root node rV, a set of demands DV with demand vD wishing to route w(v) units of flow (weight) to r, and a positive number k, the Capacitated Minimum Steiner Tree (CMStT) problem asks for a minimum Steiner tree, rooted at r, spanning the vertices in D * lrr, in which the sum of the vertex weights in every subtree connected to r is at most k. When D e V, this problem is known as the Capacitated Minimum Spanning Tree (CMST) problem. Both CMsT and CMST problems are NP-hard. In this article, we present approximation algorithms for these problems and several of their variants in network design. Our main results are the following:---We present a (³ AST + 2)-approximation algorithm for the CMStT problem, where ³ is the inverse Steiner ratio, and AST is the best achievable approximation ratio for the Steiner tree problem. Our ratio improves the current best ratio of 2AST + 2 for this problem.---In particular, we obtain (³ + 2)-approximation ratio for the CMST problem, which is an improvement over the current best ratio of 4 for this problem. For points in Euclidean and rectilinear planes, our result translates into ratios of 3.1548 and 3.5, respectively.---For instances in the plane, under the Lp norm, with the vertices in D having uniform weights, we present a nontrivial (7/5AST + 3/2)-approximation algorithm for the CMStT problem. This translates into a ratio of 2.9 for the CMST problem with uniform vertex weights in the Lpmetric plane. Our ratio of 2.9 solves the long-standing open problem of obtaining any ratio better than 3 for this case.---For the CMST problem, we show how to obtain a 2-approximation for graphs in metric spaces with unit vertex weights and k = 3,4.---For the budgeted CMST problem, in which the weights of the subtrees connected to r could be up to ± k instead of k (± e 1), we obtain a ratio of ³ p 2/±.