Stability Implies Computational Tractability: Locating a Tree in a Stable Network is Easy

In this work, we answer an open problem in the study of phylogenetic networks. Phylogenetic trees are rooted binary trees in which all edges are directed away from the root, whereas phylogenetic networks are rooted acyclic digraphs. For the purpose of evolutionary model validation, biologists often want to know whether or not a phylogenetic tree is contained in a phylogenetic network. The tree containment problem is NP-complete even for very restricted classes of networks such as tree-sibling phylogenetic networks. We prove that this problem is solvable in cubic time for stable phylogenetic networks. A linear time algorithm is also presented for the cluster containment problem.

[1]  L. Nakhleh,et al.  Computational approaches to species phylogeny inference and gene tree reconciliation. , 2013, Trends in ecology & evolution.

[2]  Daniel H. Huson,et al.  Beyond Galled Trees - Decomposition and Computation of Galled Networks , 2007, RECOMB.

[3]  Tandy J. Warnow,et al.  Phylogenetic networks: modeling, reconstructibility, and accuracy , 2004, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[4]  Manuel Spannagl,et al.  Ancient hybridizations among the ancestral genomes of bread wheat , 2014, Science.

[5]  Gabriel Cardona,et al.  Comparison of Tree-Child Phylogenetic Networks , 2007, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[6]  K. McBreen,et al.  Reconstructing reticulate evolutionary histories of plants. , 2006, Trends in plant science.

[7]  Steven Kelk,et al.  Phylogenetic Networks: Concepts, Algorithms and Applications , 2012 .

[8]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[9]  E. Rocha,et al.  Horizontal Transfer, Not Duplication, Drives the Expansion of Protein Families in Prokaryotes , 2011, PLoS genetics.

[10]  Doolittle Wf.  Phylogenetic Classification and the Universal Tree , 1999 .

[11]  Dan Gusfield,et al.  ReCombinatorics: The Algorithmics of Ancestral Recombination Graphs and Explicit Phylogenetic Networks , 2014 .

[12]  Simone Linz,et al.  Phylogenetic Networks that Display a Tree Twice , 2014, Bulletin of mathematical biology.

[13]  Louxin Zhang,et al.  Solving the Tree Containment Problem for Genetically Stable Networks in Quadratic Time , 2015, IWOCA.

[14]  Tal Dagan,et al.  Modular networks and cumulative impact of lateral transfer in prokaryote genome evolution , 2008, Proceedings of the National Academy of Sciences.

[15]  Simone Linz,et al.  Counting Trees in a Phylogenetic Network Is \#P-Complete , 2013, SIAM J. Comput..

[16]  Éva Tardos,et al.  Algorithm design , 2005 .

[17]  L. Orgel,et al.  Phylogenetic Classification and the Universal Tree , 1999 .

[18]  Steven Skiena,et al.  Lowest common ancestors in trees and directed acyclic graphs , 2005, J. Algorithms.

[19]  Kaizhong Zhang,et al.  Perfect Phylogenetic Networks with Recombination , 2001, J. Comput. Biol..

[20]  Ge Xia,et al.  Seeing the trees and their branches in the network is hard , 2007, Theor. Comput. Sci..

[21]  G. Carlsson,et al.  Topology of viral evolution , 2013, Proceedings of the National Academy of Sciences.

[22]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[23]  Leo van Iersel,et al.  Locating a tree in a phylogenetic network , 2010, Inf. Process. Lett..

[24]  Louxin Zhang,et al.  Locating a Tree in a Phylogenetic Network in Quadratic Time , 2015, RECOMB.