Print Email Facebook Twitter Hybridization Number on Three Rooted Binary Trees is EPT Title Hybridization Number on Three Rooted Binary Trees is EPT Author van Iersel, L.J.J. (TU Delft Optimization) Kelk, Steven (Universiteit Maastricht) Lekić, Nela (Universiteit Maastricht) Whidden, Chris (Fred Hutchinson Cancer Research Center) Zeh, Norbert (Dalhousie University) Date 2016 Abstract Phylogenetic networks are leaf-labeled directed acyclic graphs that are used to describe nontreelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all in-degrees minus the number of nodes plus one. The hybridization number problem takes as input a collection of rooted binary phylogenetic trees and asks to construct a phylogenetic network that contains an embedding of each of the input trees and has the smallest possible hybridization number. We present an algorithm for the hybridization number problem on three binary phylogenetic trees on n leaves that runs in time O(ckpoly(n)) with k the hybridization number of an optimal network and c some (astronomical) constant. For the case of two trees, an algorithm with running time O(3.18kn) was proposed before, whereas an algorithm with running time O(ckpoly(n)), also called an EPT algorithm, had prior to this article remained elusive for more than two trees. The algorithm for two trees uses the close connection to acyclic agreement forests to achieve a linear exponent in the running time, while previous algorithms for more than two trees (explicitly or implicitly) relied on a brute force search through all possible underlying network topologies, leading to running times that are not O(ckpoly(n)) forany c. The connection to acyclic agreement forests is much weaker for more than two trees, so even given the right agreement forest, the reconstruction of the network poses major challenges. We prove novel structural results that allow us to reconstruct a network without having to guess the underlying topology. Our techniques generalize to more than three input trees with the exception of one key lemma that maps nodes in the network to tree nodes in order to minimize the amount of guessing involved in constructing the network. The main open problem therefore is to prove results that establish such a mapping for more than three trees. Subject hybridization numberrooted phylogenetic treerooted phylogenetic networkreticulate evolutionagreement forestfixed parameter tractability To reference this document use: http://resolver.tudelft.nl/uuid:b42f38ed-f026-40f5-9431-2492791394e9 DOI https://doi.org/10.1137/15M1036579 ISSN 0895-4801 Source SIAM Journal on Discrete Mathematics, 30 (3), 1607-1631 Part of collection Institutional Repository Document type journal article Rights © 2016 L.J.J. van Iersel, Steven Kelk, Nela Lekić, Chris Whidden, Norbert Zeh Files PDF 10235106.pdf 766.48 KB Close viewer /islandora/object/uuid:b42f38ed-f026-40f5-9431-2492791394e9/datastream/OBJ/view