Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets

Sudipto Guha*, Samir Khuller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

137 Scopus citations

Abstract

In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on "spider decompositions" was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 In k, where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1 - o(1)) In k, assuming that NP ⊈ DTIME[no(log log n)], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of In k. For the weighted case we develop a new decomposition theorem and generalize the notion of "spiders" to "branch-spiders" that are used to design a new algorithm with a worst case approximation factor of 1.5 In k. We then generalize the method to yield an approximation factor of (1.35+∈) In k, for any constant ∈>0. These algorithms, although polynomial, are not very practical due to their high running time, since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 In k. The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0-1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 In n by Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35+∈) In n for any fixed ∈>0.

Original languageEnglish (US)
Pages (from-to)57-74
Number of pages18
JournalInformation and Computation
Volume150
Issue number1
DOIs
StatePublished - Apr 10 1999

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Information Systems
  • Computer Science Applications
  • Computational Theory and Mathematics

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