Abstract

ABSTRACT. We study single-sink network design problems in undirected graphs with vertex connectivity requirements. The input to these problems is an edge-weighted undirected graph G = (V,E), a sink/root vertex r, a set of terminals T ⊆ V, and integer k. The goal is to connect each terminal t ∈ T to r via k vertex-disjoint paths. In the connectivity problem, the objective is to find a min-cost subgraph of G that contains the desired paths. There is a 2-approximation for this problem when k ≤ 2 [9] but for k ≥ 3, the first non-trivial approximation was obtained in the recent work of Chakraborty, Chuzhoy and Khanna [4]; they describe and analyze an algorithm with an approximation ratio ofO(k O(k2) log 4 n) where n = |V|. In this paper, inspired by the results and ideas in [4], we show an O(k O(k) log |T|)-approximation bound for a simple greedy algorithm. Our analysis is based on the dual of a natural linear program and is of independent technical interest. We use the insights from this analysis to obtain an O(k O(k) log |T|)-approximationforthemoregeneralsingle-sinkrent-or-buynetworkdesignproblem with vertex connectivity requirements. We further extend the ideas to obtain a poly-logarithmic approximation for the single-sink buy-at-bulk problem when k = 2 and the number of cable-types is a fixed constant; we believe that this should extend to any fixed k. We also show that for the nonuniform buy-at-bulk problem, for each fixed k, a small variant of a simple algorithm suggested by Charikar and Kargiazova [5] for the case of k = 1 gives an 2 O( √ log |T|) approximation for larger k. These results show that for each of these problems, simple and natural algorithms that have been developed for k = 1have good performance forsmall k> 1. 1

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