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Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

Abstract

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question.

Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns \(\lfloor{{k}\over{2}}-1\rfloor\) element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.

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Author information

Authors and Affiliations

  1. Dept. of Comb.& Opt., U. Waterloo, Waterloo, ON, Canada, N2L 3G1

    Ashkan Aazami & Joseph Cheriyan

  2. Dept. of Comp. Sci., U. Waterloo, Waterloo, ON, Canada, N2L 3G1

    Krishnam Raju Jampani

Authors
  1. Ashkan Aazami
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  2. Joseph Cheriyan
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  3. Krishnam Raju Jampani
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Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2009 Springer-Verlag Berlin Heidelberg

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Aazami, A., Cheriyan, J., Jampani, K.R. (2009). Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_1

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  • DOI: https://doi.org/10.1007/978-3-642-03685-9_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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