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

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Automata, Languages and Programming (ICALP 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5555))

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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.

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Author information

Authors and Affiliations

  1. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA, 02139, USA

    Erik D. Demaine

  2. AT&T Labs — Research, 180 Park Ave., Florham Park, NJ, 07932, USA

    MohammadTaghi Hajiaghayi

  3. Department of Computer Science, Brown University, Providence, RI, 02912, USA

    Philip N. Klein

Authors
  1. Erik D. Demaine
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  2. MohammadTaghi Hajiaghayi
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  3. Philip N. Klein
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, Georges Köhler Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Department of Computer and Systems Sciences, Sapienza University of Rome, Via Ariosto 25, 00184, Roma, Italy

    Alberto Marchetti-Spaccamela

  3. Google R&D Center, School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Yossi Matias

  4. University of Patras and CTI, N. Kazantzaki Street 1, 26504 Rion, Patras, Greece

    Sotiris Nikoletseas

  5. RWTH Aachen, Lehrstuhl Informatik 7, Ahornstraße 55, 52056, Aachen, Germany

    Wolfgang Thomas

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© 2009 Springer-Verlag Berlin Heidelberg

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Demaine, E.D., Hajiaghayi, M., Klein, P.N. (2009). Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_28

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  • DOI: https://doi.org/10.1007/978-3-642-02927-1_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02926-4

  • Online ISBN: 978-3-642-02927-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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