Algorithmica

, Volume 71, Issue 4, pp 797–811 | Cite as

Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem

Article

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 \(\frac{7}{4}\) 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 \(\frac{5}{3}\)-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. Extending 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 analyses are tight.

Keywords

Approximation algorithm Spanning tree 

Notes

Acknowledgements

We thank Hartmut Noltemeier and Hans-Christoph Wirth for helpful discussions and the anonymous reviewers for helpful suggestions to improve the paper.

References

  1. 1.
    Binkele-Raible, D., Fernau, H., Gaspers, S., Liedloff, M.: Exact and parameterized algorithms for max internal spanning tree. Algorithmica 65(1), 95–128 (2013) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Chimani, M., Spoerhase, J.: Approximating spanning trees with few branches. In: Proc. 10th International Workshop on Approximation and Online Algorithms (WAOA’12), pp. 30–41 (2012) Google Scholar
  3. 3.
    Flandrin, E., Kaiser, T., Kuzel, R., Li, H., Ryjácek, Z.: Neighborhood unions and extremal spanning trees. Discrete Math. 308(12), 2343–2350 (2008) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Fomin, F.V., Gaspers, S., Saurabh, S., Thomassé, S.: A linear vertex kernel for maximum internal spanning tree. J. Comput. Syst. Sci. 79(1), 1–6 (2013) CrossRefMATHGoogle Scholar
  5. 5.
    Knauer, M., Spoerhase, J.: Better approximation algorithms for the maximum internal spanning tree problem. In: Proc. 11th Algorithms and Data Structures Symposium (WADS’09), pp. 459–470 (2009) CrossRefGoogle Scholar
  6. 6.
    Lu, H.I., Ravi, R.: The power of local optimization: approximation algorithms for maximum-leaf spanning tree. In: Proc. 30th Annual Allerton Conference on Communication, Control and Computing, pp. 533–542 (1992) Google Scholar
  7. 7.
    Orlin, J.B., Punnen, A.P., Schulz, A.S.: Approximate local search in combinatorial optimization. SIAM J. Comput. 33(5), 1201–1214 (2004) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Prieto, E., Sloper, C.: Reducing to independent set structure—the case of k-internal spanning tree. Nord. J. Comput. 12(3), 308–318 (2005) MATHMathSciNetGoogle Scholar
  9. 9.
    Salamon, G.: Approximating the maximum internal spanning tree problem. Theor. Comput. Sci. 410(50), 5273–5284 (2009) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Salamon, G.: A survey on algorithms for the maximum internal spanning tree and related problems. Electron. Notes Discrete Math. 36, 1209–1216 (2010) CrossRefGoogle Scholar
  11. 11.
    Salamon, G., Wiener, G.: On finding spanning trees with few leaves. Inf. Process. Lett. 105, 164–169 (2008) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Chair of Computer Science IUniversity of WürzburgWürzburgGermany

Personalised recommendations