Approximation Algorithms for the Capacitated Minimum Spanning Tree Problem and Its Variants in Network Design

Abstract

Given an undirected graph G=(V,E) with non-negative costs on its edges, a root node r ∈ V, a set of demands D ⊆ V with demand v ∈ D 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∪{r}, in which the sum of the vertex weights in every subtree hanging off r is at most k. When D=V, this problem is known as the Capacitated Minimum Spanning Tree (CMST) problem. Both CMStT and CMST problems are NP-hard. In this paper, we present approximation algorithms for these problems and several of their variants in network design. Our main results are the following.