Abstract
In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G = (V,E), a rational number k ≥ 1 and a weight function \(w: V \longmapsto Q_{\geq 1}\) on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G,w, k 〉 of Weighted Max Leaf in linear time into an equivalent instance 〈G′,w′, k′ 〉 such that |V′| ≤ 5.5k′ and k′ ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G that excludes some simple substructures always contains a spanning tree with at least |V|/5.5 leaves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Böcker, S., Briesemeister, S., Bui, Q.B., Truss, A.: A fixed-parameter approach for weighted cluster editing. In: Series on Advances in Bioinformatics and Computational Biology, vol. 5, pp. 211–220. Imperial College Press, London (2008)
Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Proc. 4th IWPEC, pp. 17–37 (2009)
Bonsma, P.S., Zickfeld, F.: Spanning trees with many leaves in graphs without diamonds and blossoms. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 531–543. Springer, Heidelberg (2008)
Crescenzi, P., Silvestri, R., Trevisan, L.: On weighted vs unweighted versions of combinatorial optimization problems. Inf. Comput. 167(1), 10–26 (2001)
Daligault, J., Thomassé, S.: On finding directed trees with many leaves. In: Proc. 4th IWPEC, pp. 86–97 (2009)
Estivill-Castro, V., Fellows, M., Langston, M., Rosamond, F.: FPT is P-time extremal structure I. In: Proc. 1st ACiD, pp. 1–41 (2005)
Fernau, H., Kneis, J., Kratsch, D., Langer, A., Liedloff, M., Raible, D., Rossmanith, P.: An exact algorithm for the maximum leaf spanning tree problem. In: Proc. 4th IWPEC, pp. 161–172 (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science). An EATCS Series. Springer, New York (2006)
Fujito, T., Nagamochi, H.: A 2-approximation algorithm for the minimum weight edge dominating set problem. Discrete Appl. Math. 118(3), 199–207 (2002)
Griggs, J.R., Kleitman, D., Shastri, A.: Spanning trees with many leaves in cubic graphs. J. Graph Theory 13, 669–695 (1989)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
Jansen, B.: Kernelization for maximum leaf spanning tree with positive vertex weights. Technical Report UU-CS-2009-027, Department of Information and Computing Sciences, Utrecht University (2009), http://www.cs.uu.nl/research/techreps/UU-CS-2009-027.html
Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J. Discret. Math. 4(1), 99–106 (1991)
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)
Kratsch, S.: Polynomial kernelizations for MIN \(F^+ \Pi_1\) and MAX NP. In: Albers, S., Marion, J.-Y. (eds.) 26th STACS, Leibniz-Zentrum fuer Informatik (March 2009)
Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 63–77 (2003)
Raible, D., Fernau, H.: An amortized search tree analysis for k-leaf spanning tree. In: Proc. 36th SOFSEM, pp. 672–684 (2010)
Solis-oba, R.: 2-approximation algorithm for finding a spanning tree with the 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. Information Processing Letters 13(1), 8–11 (1981)
Zimand, M.: Weighted NP optimization problems: Logical definability and approximation properties. SIAM J. Comput. 28(1), 36–56 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jansen, B. (2010). Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights. In: Calamoneri, T., Diaz, J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13073-1_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-13073-1_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13072-4
Online ISBN: 978-3-642-13073-1
eBook Packages: Computer ScienceComputer Science (R0)