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Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

  • Erik D. Demaine
  • MohammadTaghi Hajiaghayi
  • Philip N. Klein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5555)

Abstract

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ(logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games.

The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 n), or O(log2 n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group.

Keywords

Approximation Algorithm Planar Graph Steiner Tree Network Design Problem Steiner Tree Problem 
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 2009

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • MohammadTaghi Hajiaghayi
    • 2
  • Philip N. Klein
    • 3
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.AT&T Labs — ResearchFlorham ParkUSA
  3. 3.Department of Computer ScienceBrown UniversityProvidenceUSA

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