an:06620945
Zbl 1348.68068
Kelk, Steven; Fischer, Mareike; Moulton, Vincent; Wu, Taoyang
Reduction rules for the maximum parsimony distance on phylogenetic trees
EN
Theor. Comput. Sci. 646, 1-15 (2016).
00358560
2016
j
68Q25 05C05 92D15
phylogenetics; parsimony; fixed parameter tractability; chain; incongruence; treewidth
Summary: 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 \textsc{tbr} and \textsc{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 \textsc{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 \textsc{tbr} and \textsc{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 \textsc{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.