Skip to main content
SpringerLink
Log in

A 2-Approximation Algorithm for Finding a Spanning Tree with Maximum Number of Leaves

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study the problem of finding a spanning tree with maximum number of leaves. We present a simple, linear time 2-approximation algorithm for this problem, improving on the previous best known algorithm for the problem, which has approximation ratio 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alon, N., Fomin, F.V., Gutin, G., Krivelevich, M., Saurabh, S.: Spanning directed trees with many leaves. SIAM J. Discret. Math. 23(1), 466–476 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Binkele-Raible, D., Fernau, H.: An exact exponential-time algorithm for the directed maximum leaf spanning tree problem. J. Discret. Algorithms 15, 43–55 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Binkele-Raible, D., Fernau, H., Fomin, F.V., Lokshtanov, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: on out-trees with many leaves. ACM Trans. Algorithms 8(4) (2012) (Article 38)

  4. Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonsma, P.: Max-leaves spanning tree is APX-hard for cubic graphs. J. Discret. Algorithms 12, 14–23 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonsma, P., Brueggemann, T., Woeginger, G.J.: A faster FPT algorithm for finding spanning trees with many leaves. In: Mathematical Foundations of Computer Science (MFCS 2003). Volume 2747 of Lecture Notes in Computer Science, pp. 259–268. Springer, Berlin (2003)

  7. Bonsma, P., Dorn, F.: Tight bounds and a fast FPT algorithm for directed max-leaf spanning tree. ACM Trans. Algorithms 7(4) (2011) (Article 44)

  8. Bonsma, P., Zickfeld, F.: A 3/2-approximation algorithm for finding spanning trees with many leaves in cubic graphs. SIAM J. Discret. Math. 25(4), 1652–1666 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonsma, P., Zickfeld, F.: Improved bounds for spanning trees with many leaves. Discret. Math. 312(6), 1178–1194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Correa, J.R., Fernandes, C.G., Matamala, M., Wakabayashi, Y.: A 5/3-approximation for finding spanning trees with many leaves in cubic graphs. In: Approximation and Online Algorithms (WAOA 2007). Volume 4927 of Lecture Notes in Computer Science, pp. 184–192. Springer, Berlin (2008)

  11. Daligault, J., Gutin, G., Kim, E.J., Yeo, A.: FPT algorithms and kernels for the directed \(k\)-leaf problem. J. Comput. Syst. Sci. 76(2), 144–152 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daligault, J., Thomassé, S.: On finding directed trees with many leaves. In: Parameterized and Exact Computation (IWPEC 2009). Volume 5917 of Lecture Notes in Computer Science, pp. 86–97. Springer, Berlin (2009)

  13. Drescher, M., Vetta, A.: An approximation algorithm for the max leaf spanning arborescence problem. ACM Trans. Algorithms 6(3) (2010) (Article 46)

  14. Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-time extremal structure I. In: Algorithms and Complexity in Durham (ACiD 2005). Volume 4 of Texts in Algorithmics, pp. 1–41. King’s College, London (2005)

  15. Fellows, M., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F., Saurabh, S.: The complexity ecology of parameters: an illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fernau, H., Kneis, J., Kratsch, D., Langer, A., Liedloff, M., Raible, D., Rossmanith, P.: An exact algorithm for the maximum leaf spanning tree problem. Theor. Comput. Sci. 412(45), 6290–6302 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujie, T.: The maximum-leaf spanning tree problem: formulations and facets. Networks 43(4), 212–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Galbiati, G., Maffioli, F., Morzenti, A.: A short note on the approximability of the maximum leaves spanning tree problem. Inf. Process. Lett. 52(1), 45–49 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Griggs, J.R., Kleitman, D.J., Shastri, A.: Spanning trees with many leaves in cubic graphs. J. Graph Theory 13(6), 669–695 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Griggs, J.R., Wu, M.: Spanning trees in graphs of minimum degree 4 or 5. Discret. Math. 104(2), 167–183 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jansen, B.M.P.: Kernelization for maximum leaf spanning tree with positive vertex weights. J. Graph Algorithms Appl. 16(4), 811–846 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamei, S., Kakugawa, H., Devismes, S., Tixeuil, S.: A self-stabilizing 3-approximation for the maximum leaf spanning tree problem in arbitrary networks. J. Comb. Optim. 25(3), 430–459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J. Discret. Math. 4(1), 99–106 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Loryś, K., Zwoźniak, G.: Approximation algorithm for the maximum leaf spanning tree problem for cubic graphs. In: Algorithms-ESA 2002. Volume 2461 of Lecture Notes in Computer Science, pp. 686–697. Springer, Berlin (2002)

  26. Lu, H., Ravi, R.: The power of local optimization: Approximation algorithms for maximum-leaf spanning tree. In: Proceedings of the Thirtieth Annual Allerton Conference on Communication, Control and Computing, pp. 533–542 (1992)

  27. Lu, H., Ravi, R.: Approximating maximum leaf spanning trees in almost linear time. J. Algorithms 29(1), 132–141 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Payan, C., Tchuente, M., Xuong, N.H.: Arbres avec un nombre maximum de sommets pendants. Discret. Math. 49(3), 267–273 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Raible, D., Fernau, H.: An amortized search tree analysis for k-leaf spanning tree. In: SOFSEM 2010: Theory and Practice of Computer Science. Volume 5901 of Lecture Notes in Computer Science, pp. 672–684. Springer, Berlin (2010)

  30. Ruan, L., Du, H., Jia, X., Wu, W., Li, Y., Ko, K.: A greedy approximation for minimum connected dominating sets. Theor. Comput. Sci. 329(1–3), 325–330 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schwartges, N., Spoerhase, J., Wolff, A.: Approximation algorithms for the maximum leaf spanning tree problem on acyclic digraphs. In: Approximation and Online Algorithms (WAOA 2011). Volume 7164 of Lecture Notes in Computer Science, pp. 77–88. Springer, Berlin (2012)

  32. Solis-Oba, R.: 2-approximation algorithm for finding a spanning tree with maximum number of leaves. In: Algorithms-ESA 1998. Volume 1461 of Lecture Notes in Computer Science, pp. 441–452. Springer, Berlin (1998)

  33. Solis-Oba, R.: 2-Approximation Algorithm for Finding a Spanning Tree with Maximum Number of Leaves. Technical report TR 98-1-010. Max Planck Institute for Computer Science, Saarbruecken (1998). http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1998-1-010

  34. Storer, J.A.: Constructing full spanning trees for cubic graphs. Inf. Process. Lett. 13(1), 8–11 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Western Ontario, London, ON, N6A 5B7, Canada

    Roberto Solis-Oba

  2. Faculty of EEMCS, University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands

    Paul Bonsma

  3. Mathematics Department, Humboldt University Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Stefanie Lowski

Authors
  1. Roberto Solis-Oba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Bonsma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefanie Lowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Solis-Oba.

Additional information

An extended abstract of this paper appeared in the proceedings of ESA 1998 [32].

Roberto Solis-Oba: Research of this author partially supported by Grant 227829-2009 from the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Solis-Oba, R., Bonsma, P. & Lowski, S. A 2-Approximation Algorithm for Finding a Spanning Tree with Maximum Number of Leaves. Algorithmica 77, 374–388 (2017). https://doi.org/10.1007/s00453-015-0080-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-015-0080-0

Keywords

Search

Navigation