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The k-Leaf Spanning Tree Problem Admits a Klam Value of 39

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Book cover Combinatorial Algorithms (IWOCA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9538))

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Abstract

The klam value of an algorithm that runs in time \(O^*(f(k))\) is the maximal value k such that \(f(k)<10^{20}\). Given a graph G and a parameter k, the k -Leaf Spanning Tree ( k -LST) problem asks if G contains a spanning tree with at least k leaves. This problem has been extensively studied over the past three decades. In 2000, Fellows et al. [FSTTCS’00] asked whether it admits a klam value of 50. A steady progress towards an affirmative answer continued until 5 years ago, when an algorithm of klam value 37 was discovered. Our contribution is twofold. First, we present an \(O^*(3.188^k)\)-time parameterized algorithm for k -LST, which shows that the problem admits a klam value of 39. Second, we rely on an application of the bounded search trees technique where the correctness of rules crucially depends on the history of previously applied rules in a non-standard manner, encapsulated in a “dependency claim”. Similar claims may be used to capture the essence of other complex algorithms in a compact, useful manner.

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Notes

  1. 1.

    A connected dominating set is a subset \(U\subseteq V\) such that the subgraph of G induced by U is connected and every vertex in \(V{\setminus }U\) is a neighbor of a vertex in U.

  2. 2.

    Problematic vertices in whose examination we cannot rely on such a fact will be handled by a “marking” approach—we will be able to consider our treatment of them as better than it is, since we previously considered the treatment of the vertices that marked them as worse than it is.

  3. 3.

    For example, if \(T=(\{r,p,v,u,a,b,c\},\{(r,p),(p,v),(p,u),(r,a),(a,b),(b,c)\})\), and exactly before (after) p was inserted to T as an internal vertex, the tree was \(T_1=(\{r,p,a\},\{(r,p),(r,a)\})\) (\(T_2=(\{r,p,v,u,a,b\},\{(r,p),(p,v),(p,u),(r,a),(a,b)\})\)), then \(T'=T_2\) and \(\widetilde{T}=(\{r,p,a,b\},\{(r,p),(r,a),(a,b)\})\).

  4. 4.

    In this context, recall that we need to avoid marking vertices when it is possible, since each marked vertex increases the measure.

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Authors and Affiliations

  1. Department of Computer Science, Technion IIT, 32000, Haifa, Israel

    Meirav Zehavi

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  1. Meirav Zehavi
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Correspondence to Meirav Zehavi .

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Editors and Affiliations

  1. Dept. of Computer Science, University of Verona, Verona, Italy

    Zsuzsanna Lipták

  2. Dept. of Computing and Software, McMaster University, Hamilton, Canada

    William F. Smyth

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Zehavi, M. (2016). The k-Leaf Spanning Tree Problem Admits a Klam Value of 39. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_29

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  • DOI: https://doi.org/10.1007/978-3-319-29516-9_29

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