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Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees

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Abstract

We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise naturally as a result of statistical uncertainty in current tree construction methods. The size of an MAF corresponds to the subtree prune-and-regraft distance of the two trees and is intimately connected to their hybridization number. These distance measures are essential tools for understanding reticulate evolution, such as lateral gene transfer, recombination, and hybridization. Our algorithms nearly match the running times of the currently best algorithms for the binary case. This is achieved using a combination of efficient branching rules (similar to but more complex than in the binary case) and a novel edge protection scheme that further reduces the size of the search space the algorithms need to explore.

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Notes

  1. This is similar to the generalization of the hybridization number used by Linz and Semple [18].

  2. In fact, using the same ideas as in the proof of Lemma 3, it is not difficult to see that this expansion never precludes obtaining the same forest \(F \div E\) by cutting a different set of \(|E|\) edges. We discuss the importance of this to hybridization and reticulate analysis in Sect. 6.

  3. We excluded this case from the statement of the lemma, in order to keep the cases covered by the different lemmas disjoint, but the lemma also holds for \(s = 1\). A similar comment applies to Lemma 12.

  4. This is a fairly loose bound on \(I(k,t)\), but it is easy to manipulate.

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Authors and Affiliations

  1. Program in Computational Biology, Fred Hutchinson Cancer Research Center, Seattle, WA, USA

    Chris Whidden

  2. Faculty of Computer Science, Dalhousie University, Halifax, NS, Canada

    Robert G. Beiko & Norbert Zeh

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  1. Chris Whidden
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  2. Robert G. Beiko
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  3. Norbert Zeh
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Correspondence to Norbert Zeh.

Additional information

This work was done while Chris Whidden was a PhD student at Dalhousie University. Chris Whidden was supported by a Killam predoctoral scholarship and the Tula Foundation, Robert G. Beiko is a Canada Research Chair and was supported by NSERC, Genome Atlantic, and the Canada Foundation for Innovation, and Norbert Zeh is a Canada Research Chair and was supported by NSERC and the Canada Foundation for Innovation.

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Whidden, C., Beiko, R.G. & Zeh, N. Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees. Algorithmica 74, 1019–1054 (2016). https://doi.org/10.1007/s00453-015-9983-z

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