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Approximate k-Steiner Forests Via the Lagrangian Relaxation Technique with Internal Preprocessing

  • Danny Segev
  • Gil Segev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

Abstract

An instance of the k-Steiner forest problem consists of an undirected graph G = (V,E), the edges of which are associated with non-negative costs, and a collection \({\cal D} = \{ (s_i, t_i) : 1 \leq i \leq d \}\) of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest \({\cal F} \subseteq G\) connects a demand (s i , t i ) when it contains an s i -t i path. Given a requirement parameter \(k \leq |{\cal D}|\), the goal is to find a minimum cost forest that connects at least k demands in \({\cal D}\). This problem has recently been studied by Hajiaghayi and Jain [SODA ’06], whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research.

In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of \(O( {\rm min} \{ n^{ 2/3 }, \sqrt{d} \} \cdot \log d )\) of optimal, where n is the number of vertices in the input graph and d is the number of demands.

Keywords

Steiner Tree Lagrangian Relaxation Input Graph Forest Problem Steiner Forest 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Danny Segev
    • 1
  • Gil Segev
    • 2
  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael
  2. 2.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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