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Spotting Trees with Few Leaves

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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Abstract

We show two results related to the Hamiltonicity and \(k\) -Path algorithms in undirected graphs by Björklund [FOCS’10], and Björklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some \(k\)-vertex tree with \(l\) leaves in an \(n\)-vertex undirected graph in \(O^*(1.657^k2^{l/2})\) time. It can be applied as a subroutine to solve the \(k\) -Internal Spanning Tree (\(k\)-IST) problem in \(O^*({\text {min}}(3.455^k, 1.946^n))\) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time, we break the natural barrier of \(O^*(2^n)\). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for \(k\) -Path and Hamiltonicity in any graph of maximum degree \(\Delta =4,\ldots ,12\) or with vector chromatic number at most \(8\).

Work partially supported by the National Science Centre of Poland, grant number 2013/09/B/ST6/03136 and ERC StG project PAAl no. 259515 (ŁK). The paper was prepared while the second author held a post-doctoral position at Warsaw Center of Mathematics and Computer Science.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Lund University, Lund, Sweden

    Andreas Björklund

  2. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Vikram Kamat & Łukasz Kowalik

  3. Department of Computer Science, Technion – Israel Institute of Technology, Haifa, Israel

    Meirav Zehavi

Authors
  1. Andreas Björklund
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  2. Vikram Kamat
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  3. Łukasz Kowalik
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  4. Meirav Zehavi
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Corresponding author

Correspondence to Łukasz Kowalik .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Björklund, A., Kamat, V., Kowalik, Ł., Zehavi, M. (2015). Spotting Trees with Few Leaves. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_20

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_20

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