Abstract
We consider the parameterized version of the maximum internal spanning tree problem: given an n-vertex graph and a parameter k, does the graph have a spanning tree with at least k internal vertices? Fomin et al. [J. Comput. System Sci., 79:1–6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a 3k-vertex kernel for this problem. Here we propose a novel way to use the same reduction rule, resulting in an improved 2k-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a \(4^k \cdot n^{O(1)}\)-time deterministic algorithm, improving all previous algorithms for the problem.
Supported by the National Natural Science Foundation of China under grants 61232001, 61472449, and 61420106009.
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Li, W., Wang, J., Chen, J., Cao, Y. (2015). A 2k-vertex Kernel for Maximum Internal Spanning Tree. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_41
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DOI: https://doi.org/10.1007/978-3-319-21840-3_41
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