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Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem

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Algorithms and Data Structures (WADS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5664))

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Abstract

We examine the problem of determining a spanning tree of a given graph such that the number of internal nodes is maximum. The best approximation algorithm known so far for this problem is due to Prieto and Sloper and has a ratio of 2. For graphs without pendant nodes, Salamon has lowered this factor to \(\frac74\) by means of local search. However, the approximative behaviour of his algorithm on general graphs has remained open. In this paper we show that a simplified and faster version of Salamon’s algorithm yields a \(\frac53\)-approximation even on general graphs. In addition to this, we investigate a node weighted variant of the problem for which Salamon achieved a ratio of 2·Δ(G) − 3. Modifying Salamon’s approach we obtain a factor of 3 + ε for any ε> 0. We complement our results with worst case instances showing that our bounds are tight.

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References

  1. Salamon, G., Wiener, G.: On finding spanning trees with few leaves. Information Processing Letters 105, 164–169 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lu, H., Ravi, R.: The power of local optimization: approximation algorithms for maximum-leaf spanning tree. Technical report, Department of Computer Science, Brown University (1996)

    Google Scholar 

  3. Flandrin, E., Kaiser, T., Kuzel, R., Li, H., Ryjcek, Z.: Neighborhood unions and extremal spanning trees. Discrete Mathematics 308(12), 2343–2350 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Prieto, E., Sloper, C.: Either/or: Using vertex cover structure in designing fpt-algorithms - the case of k-internal spanning tree. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 474–483. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  5. Prieto, E., Sloper, C.: Reducing to independent set structure – the case of k-internal spanning tree. Nord. J. Comput. 12(3), 308–318 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Fernau, H., Raible, D., Gaspers, S., Stepanov, A.A.: Exact exponential time algorithms for max internal spanning tree. In: CoRR (2008) abs/0811.1875

    Google Scholar 

  7. Salamon, G.: Approximation algorithms for the maximum internal spanning tree problem. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 90–102. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  8. Orlin, J.B., Punnen, A.P., Schulz, A.S.: Approximate local search in combinatorial optimization. SIAM J. Comput. 33(5), 1201–1214 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Salamon, G.: Approximating the maximum internal spanning tree problem. To appear in Theoretical Computer Science (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Lehrstuhl für Informatik I, Universtität Würzburg Am Hubland, 97074, Würzburg, Germany

    Martin Knauer & Joachim Spoerhase

Authors
  1. Martin Knauer
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  2. Joachim Spoerhase
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Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina Gavrilova

  3. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, P.O. Box, K1S 5B6, Ontario, Canada

    Jörg-Rüdiger Sack

  4. University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

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© 2009 Springer-Verlag Berlin Heidelberg

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Knauer, M., Spoerhase, J. (2009). Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_40

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  • DOI: https://doi.org/10.1007/978-3-642-03367-4_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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