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Parameterized Algorithms for Directed Maximum Leaf Problems

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Automata, Languages and Programming (ICALP 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4596))

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Abstract

We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family \(\cal L\) that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in \(\cal L\). Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a ‘spanning tree with many leaves’ in the undirected case, and which is interesting on its own: If a digraph \(D\in \cal L\) of order n with minimum in-degree at least 3 contains a rooted spanning tree, then D contains one with at least (n/2)1/5− 1 leaves.

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References

  1. Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley, Chichester (2000)

    MATH  Google Scholar 

  2. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, Heidelberg (2000)

    Google Scholar 

  3. Bodlaender, H.L.: On linear time minor tests and depth-first search. Journal of Algorithms 14, 1–23 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bodlaender, H.L.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonsma, P.S., Brueggermann, 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)

    Google Scholar 

  6. Cesati, M.: Compendium of parameterized problems (September 2006), http://bravo.ce.uniroma2.it/home/cesati/research/compendium.pdf

  7. Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-Time Extremal Structure I. In: Proc. ACiD, pp. 1–41 (2005)

    Google Scholar 

  8. Courcelle, B.: The Monadic second-order logic of graphs I: recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. Informatique Théorique et Applications (ITA) 26, 257–286 (1992)

    MATH  MathSciNet  Google Scholar 

  10. Ding, G., Johnson, T., Seymour, P.: Spanning trees with many leaves. Journal of Graph Theory 37, 189–197 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)

    Google Scholar 

  12. Fellows, M.R., Langston, M.A.: On well-partial-order theory and its applications to combinatorial problems of VLSI design. SIAM Journal on Discrete Mathematics 5, 117–126 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fellows, M.R., McCartin, C., Rosamond, F.A., Stege, U.: Coordinated 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)

    Google Scholar 

  14. Fellows, M.: Private communications (2005-2006)

    Google Scholar 

  15. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)

    Google Scholar 

  16. Galbiati, G., Maffioli, F., Morzenti, A.: A short note on the approximability of the maximum leaves spanning tree problem. Information Processing Letters 52, 45–49 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Galbiati, G., Morzenti, A., Maffioli, F.: On the approximability of some maximum spanning tree problems. Theoretical Computer Science 181, 107–118 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Garey, M.R., Johnson, D.S.: Computers and Intractability. W.H. Freeman and Co, New York (1979)

    MATH  Google Scholar 

  19. Griggs, J.R., Wu, M.: Spanning trees in graphs of minimum degree four or five. Discrete Mathematics 104, 167–183 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gutin, G.: The radii of n-partite tournaments. Math. Notes 40, 743–744 (1986)

    MATH  MathSciNet  Google Scholar 

  21. Gutin, G., Yeo, A.: Some Parameterized Problems on Digraphs. To appear in The Computer Journal

    Google Scholar 

  22. Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM Journal on Discrete Mathematics 4, 99–106 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Complementary Definitions of Programming Language Semantics 42, 345–350 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and searching. Mes premieres constructions de programmes 55, 181–184 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Linial, N., Sturtevant, D.: Unpublished result (1987)

    Google Scholar 

  26. Lu, H.-I., Ravi, R.: Approximating maximum leaf spanning trees in almost linear time. Interval Mathematics 29, 132–141 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Möhring, R.H.: Graph problems related to gate matrix layout and PLA folding. In Computational Graph Theory 7(Comput. Suppl.), 17–51 (1990)

    Google Scholar 

  28. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, oxford (2006)

    MATH  Google Scholar 

  29. Petrovic, V., Thomassen, C.: Kings in k-partite tournaments. Discrete Mathematics 98, 237–238 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. 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)

    Chapter  Google Scholar 

  31. Robertson, N., Seymour, P.D.: Graph minors-a survey. In: Anderson, I. (ed.) Surveys in Combinatorics, pp. 153–171. Cambridge Univ. Press, Cambridge (1985)

    Google Scholar 

  32. Robertson, N., Seymour, P.D.: Graph minors I: Excluding a forest. Journal of Combinatorial Theory Series B 35, 39–61 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  33. Robertson, N., Seymour, P.D.: Graph minors II: Algorithmic aspects of tree-width. Journal of Algorithms 7, 309–322 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wu, B.Y., Chao, K.: Spanning Trees and Optimization Problems, CRC Press (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel

    Noga Alon & Michael Krivelevich

  2. Department of Informatics, University of Bergen, POB 7803, 5020 Bergen, Norway

    Fedor V. Fomin & Saket Saurabh

  3. Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

    Gregory Gutin

  4. The Institute of Mathematical Sciences, Chennai, 600 017, Email: saket@imsc.res.in, India

    Saket Saurabh

Authors
  1. Noga Alon
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  2. Fedor V. Fomin
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  3. Gregory Gutin
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  4. Michael Krivelevich
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  5. Saket Saurabh
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Editor information

Lars Arge Christian Cachin Tomasz Jurdziński Andrzej Tarlecki

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© 2007 Springer-Verlag Berlin Heidelberg

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Alon, N., Fomin, F.V., Gutin, G., Krivelevich, M., Saurabh, S. (2007). Parameterized Algorithms for Directed Maximum Leaf Problems. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_32

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  • DOI: https://doi.org/10.1007/978-3-540-73420-8_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73419-2

  • Online ISBN: 978-3-540-73420-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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