On the fixed parameter tractability of agreement-based phylogenetic distances
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Three important and related measures for summarizing the dissimilarity in phylogenetic trees are the minimum number of hybridization events required to fit two phylogenetic trees onto a single phylogenetic network (the hybridization number), the (rooted) subtree prune and regraft distance (the rSPR distance) and the tree bisection and reconnection distance (the TBR distance) between two phylogenetic trees. The respective problems of computing these measures are known to be NP-hard, but also fixed-parameter tractable in their respective natural parameters. This means that, while they are hard to compute in general, for cases in which a parameter (here the hybridization number and rSPR/TBR distance, respectively) is small, the problem can be solved efficiently even for large input trees. Here, we present new analyses showing that the use of the “cluster reduction” rule—already defined for the hybridization number and the rSPR distance and introduced here for the TBR distance—can transform any \(O(f(p) \cdot n)\)-time algorithm for any of these problems into an \(O(f(k) \cdot n)\)-time one, where n is the number of leaves of the phylogenetic trees, p is the natural parameter and k is a much stronger (that is, smaller) parameter: the minimum level of a phylogenetic network displaying both trees.
KeywordsPhylogenetic network Hybridization number Cluster reduction SPR distance TBR distance
Mathematics Subject Classification05C85 68R10 92-08 05C05 05C60
The third author gratefully acknowledges the scholarship supplied to her from the Republic of Turkey, Ministry of National Education. Part of the work has been conceived at the 7th workshop on Graph Classes, Optimization, and Width Parameters.
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