Reduction rules for the maximum parsimony distance on phylogenetic trees

In phylogenetics, distances are often used to measure the incongruence between a pair of phylogenetic trees that are reconstructed by different methods or using different regions of genome. Motivated by the maximum parsimony principle in tree inference, we recently introduced the maximum parsimony (MP) distance, which enjoys various attractive properties due to its connection with several other well-known tree distances, such as TBR and SPR. Here we show that computing the MP distance between two trees, a NP-hard problem in general, is fixed parameter tractable in terms of the TBR distance between the tree pair. Our approach is based on two reduction rules--the chain reduction and the subtree reduction--that are widely used in computing TBR and SPR distances. More precisely, we show that reducing chains to length 4 (but not shorter) preserves the MP distance. In addition, we describe a generalization of the subtree reduction which allows the pendant subtrees to be rooted in different places, and show that this still preserves the MP distance. On a slightly different note we also show that Monadic Second Order Logic (MSOL), posited over an auxiliary graph structure known as the display graph (obtained by merging the two trees at their leaves), can be used to obtain an alternative proof that computation of MP distance is fixed parameter tractable in terms of TBR-distance. We conclude with an extended discussion in which we focus on similarities and differences between MP distance and TBR distance and present a number of open problems. One particularly intriguing question, emerging from the MSOL formulation, is whether two trees with bounded MP distance induce display graphs of bounded treewidth.

[1]  Mike A. Steel,et al.  Refining Phylogenetic Trees Given Additional Data: An Algorithm Based on Parsimony , 2009, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[2]  Frank Harary,et al.  Graph Theory , 2016 .

[3]  Jianer Chen,et al.  Parameterized and approximation algorithms for maximum agreement forest in multifurcating trees , 2015, Theor. Comput. Sci..

[4]  W. Fitch.  Toward Defining the Course of Evolution: Minimum Change for a Specific Tree Topology , 1971 .

[5]  Julia Chuzhoy,et al.  Excluded Grid Theorem: Improved and Simplified , 2015, STOC.

[6]  M. Steel,et al.  Distributions of Tree Comparison Metrics—Some New Results , 1993 .

[7]  Steven Kelk.  A note on convex characters and Fibonacci numbers , 2015, ArXiv.

[8]  David Bryant,et al.  Compatibility of unrooted phylogenetic trees is FPT , 2006, Theor. Comput. Sci..

[9]  M. Bordewich,et al.  Computing the Hybridization Number of Two Phylogenetic Trees Is Fixed-Parameter Tractable , 2007, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[10]  Steven Kelk,et al.  On the Complexity of Computing MP Distance Between Binary Phylogenetic Trees , 2014, ArXiv.

[11]  Steven Kelk,et al.  Journal of Graph Algorithms and Applications on Low Treewidth Graphs and Supertrees , 2022 .

[12]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[13]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

[14]  M. Steel,et al.  Subtree Transfer Operations and Their Induced Metrics on Evolutionary Trees , 2001 .

[15]  Vincent Moulton,et al.  A parsimony-based metric for phylogenetic trees , 2015, Adv. Appl. Math..

[16]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[17]  Jun Li,et al.  Analysis on the reconstruction accuracy of the Fitch method for inferring ancestral states , 2011, BMC Bioinformatics.

[18]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs VIII: Orientations , 1995, Ann. Pure Appl. Log..

[19]  N. Zeh,et al.  Supertrees Based on the Subtree Prune-and-Regraft Distance , 2014, Systematic biology.

[20]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[21]  Steven Kelk,et al.  On the Maximum Parsimony Distance Between Phylogenetic Trees , 2014, Annals of Combinatorics.

[22]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[23]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[24]  Steven Kelk,et al.  A Linear Bound on the Number of States in Optimal Convex Characters for Maximum Parsimony Distance , 2015, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[25]  Leo van Iersel,et al.  Phylogenetic incongruence through the lens of Monadic Second Order logic , 2015, J. Graph Algorithms Appl..

[26]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[27]  B. Mohar,et al.  Graph Minors , 2009 .

[28]  Glenn Hickey,et al.  SPR Distance Computation for Unrooted Trees , 2008, Evolutionary bioinformatics online.

[29]  Sudheer Vakati,et al.  Compatibility, Incompatibility, Tree-Width, and Forbidden Phylogenetic Minors , 2015, Electron. Notes Discret. Math..