Abstract
An out-tree T of a directed graph D is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By ℓ(D) and ℓ s (D) we denote the maximum number of leaves over all out-trees and out-branchings of D, respectively. We give fixed parameter tractable algorithms for deciding whether ℓ s (D) ≥ k and whether ℓ(D) ≥ k for a digraph D on n vertices, both with time complexity 2O(klogk) ·n O(1). This proves the problem for out-branchings to be in FPT, and improves on the previous complexity of \(2^{O(k\log^2 k)} \cdot n^{O(1)}\) for out-trees. To obtain the algorithm for out-branchings, we prove that when all arcs of D are part of at least one out-branching, ℓ s (D) ≥ ℓ(D)/3. The second bound we prove states that for strongly connected digraphs D with minimum in-degree 3, \(\ell_s(D)\geq \Theta(\sqrt{n})\), where previously \(\ell_s(D)\geq \Theta(\sqrt[3]{n})\) was the best known bound. This bound is tight, and also holds for the larger class of digraphs with minimum in-degree 3 in which every arc is part of at least one out-branching.
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References
Alon, N., Fomin, F.V., Gutin, G., Krivelevich, M., Saurabh, S.: Better algorithms and bounds for directed maximum leaf problems. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 316–327. Springer, Heidelberg (2007), http://arxiv.org/abs/0803.0701
Alon, N., Fomin, F.V., 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)
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernet. 11, 1–21 (1993)
Bonsma, P., Brüggemann, T., Woeginger, G.J.: A faster FPT algorithm for finding spanning trees with many leaves. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 259–268. Springer, Heidelberg (2003)
Bonsma, P., Dorn, F.: An FPT algorithm for directed spanning k-leaf (2007), http://arxiv.org/abs/0711.4052
Bonsma, P., 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)
Demaine, E., Gutin, G., Marx, D., Stege, U.: 07281 Open problems – Structure theory and FPT algorithmics for graphs, digraphs and hypergraphs. In: Dagstuhl Seminar Proceedings 07281 (2007), http://drops.dagstuhl.de/opus/volltexte/2007/1254
Ding, G., Johnson, T., Seymour, P.: Spanning trees with many leaves. J. Graph Theory 37, 189–197 (2001)
Drescher, M., Vetta, A.: An approximation algorithm for the maximum leaf spanning arborescence problem (manuscript, 2007)
Fellows, M.R., McCartin, C., Rosamond, F.A., Stege, U.: Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 240–251. Springer, Heidelberg (2000)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)
Gutin, G., Razgon, I., Kim, E.J.: Minimum leaf out-branching problems. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 235–246. Springer, Heidelberg (2008)
Gutin, G., Yeo, A.: Some parameterized problems on digraphs. Comput. J. 51, 363–371 (2008)
Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J. Discrete Math. 4, 99–106 (1991)
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)
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Bonsma, P., Dorn, F. (2008). Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_19
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DOI: https://doi.org/10.1007/978-3-540-87744-8_19
Publisher Name: Springer, Berlin, Heidelberg
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