Upgrading Bottleneck Constrained Forests

Sven Oliver Krumke, Madhav V. Marathe, Hartmut Noltemeier, S. S. Ravi, Hans-Christoph Wirth

Abstract: We study bottleneck constrained network upgrading problems. We are given an edge weighted graph G=(V,E) where node v ∈ V can be upgraded at a cost of c(v). This upgrade reduces the delay of each link emanating from v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a good performance. The performance is measured by the bottleneck weight of a constrained forest defined by a proper function [GW95]. These problems are a generalization of the node weighted constrained forest problems studied by Klein and Ravi [KR95]. The main result of the paper is a polynomial time approximation algorithm for this problem with performance guarantee of <math display='block'> <mrow> <mn>2</mn><mi>ln</mi><mrow><mo>(</mo> <mrow> <msqrt> <mi>e</mi> </msqrt> <mo>/</mo><mn>2</mn><mo>&#x22C5;</mo><mrow><mo>|</mo> <mi>K</mi> <mo>|</mo></mrow> </mrow> <mo>)</mo></mrow> </mrow> </math> $2 \ln (\sqrt{e}/2\cdot \vert K\vert)$ , where K:={ v : f({v})=1 } is the set of terminals given by the proper function f. We also prove that the performance bound is tight up to small constant factors by providing a lower bound of ln ∣K∣. Our results are obtained by extending the elegant solution based decomposition technique of [KR95] for approximating node weighted constrained forest problems. The results presented here extend those in [KR95,KM + 97].