Abstract
The ROOTED MAXIMUM LEAF OUTBRANCHING problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least k leaves. We use the notion of s − t numbering studied in [19,6,20] to exhibit combinatorial bounds on the existence of spanning directed trees with many leaves. These combinatorial bounds allow us to produce a constant factor approximation algorithm for finding directed trees with many leaves, whereas the best known approximation algorithm has a \(\sqrt{OPT}\)-factor [11]. We also show that ROOTED MAXIMUM LEAF OUTBRANCHING admits an edge-quadratic kernel, improving over the vertex-cubic kernel given by Fernau et al [13].
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alon, N., Fomin, F., Gutin, G., Krivelevich, M., Saurabh, S.: Parameterized algorithms for directed maximum leaf problems. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 352–362. Springer, Heidelberg (2007)
Alon, N., Fomin, F., Gutin, G., Krivelevich, M., Saurabh, S.: Spanning directed trees with many leaves. SIAM J. Discrete Maths. 23(1), 466–476 (2009)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels (Extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)
Paul, S.: Bonsma and Frederic Dorn. An fpt algorithm for directed spanning k-leaf. abs/0711.4052 (2007)
Chen, J., Liu, Y.: On the parameterized max-leaf problems: digraphs and undirected graphs. Technical report, Department of Computer Science, Texas A& M University (2008)
Cheriyan, J., Reif, J.: Directed s-t numberings, rubber bands, and testing digraph k-vertex connectivity. Combinatorica 14(4), 435–451 (1994)
Daligault, J., Gutin, G., Kim, E.J., Yeo, A.: FPT algorithms and kernels for the Directed k-Leaf problem. To appear in Journal of Computer and System Sciences
Dijkstra, E.: Self-stabilizing systems in spite of distributed control. Commun. ACM 17(11), 643–644 (1974)
Ding, G., Johnson, T., Seymour, P.: Spanning trees with many leaves. J. Graph Theory 37(4), 189–197 (2001)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)
Drescher, M., Vetta, A.: An approximation algorithm for the maximum leaf spanning arborescence problem. To appear in ACM Transactions on Algorithms
Estivill-Castro, V., Fellows, M., Langston, M., Rosamond, F.: Fixed-parameter tractability is polynomial-time extremal structure theory i: The case of max leaf. In: Proc. of ACiD 2005 (2005)
Fernau, H., Fomin, F.V., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: On out-trees with many leaves. In: Albers, S., Marion, J.-Y. (eds.) 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), Dagstuhl, Germany. Leibniz International Proceedings in Informatics, vol. 3, pp. 421–432. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009), http://drops.dagstuhl.de/opus/volltexte/2009/1843
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fomin, F., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. Algorithmica 52(2), 153–166 (2008)
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)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
Kneis, J., Langer, A., Rossmanith, P.: A new algorithm for finding trees with many leaves. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 270–281. Springer, Heidelberg (2008)
Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Rosenstiehl, P. (ed.) Theory of Graphs: Internat. Sympos.: Rome, pp. 215–232 (1966)
Linial, N., Lovasz, L., Wigderson, A.: Rubber bands, convex embeddings and graph connectivity. Combinatorica 8, 91–102 (1988)
Niedermeier, R.: Invitation to fixed parameter algorithms. Oxford Lectures Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
Solis-Oba, R.: 2-approximation algorithm for finding a spanning tree with maximum number of leaves. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 441–452. Springer, Heidelberg (1998)
Storer, J.A.: Constructing full spanning trees for cubic graphs. Inform Process Lett. 13, 8–11 (1981)
Wu, J., Li, H.: On calculating connected dominating set for efficient routing in ad hoc wireless networks. In: DIALM 1999: Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications, pp. 7–14. ACM Press, New York (1999)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Daligault, J., Thomassé, S. (2009). On Finding Directed Trees with Many Leaves. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-11269-0_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11268-3
Online ISBN: 978-3-642-11269-0
eBook Packages: Computer ScienceComputer Science (R0)