Computing phylogenetic roots with bounded degrees and errors

Abstract

Given a set of species and their similarity data, an important problem in evolutionary biology is how to reconstruct a phylogeny (also called evolutionary tree) so that species are close in the phylogeny if and only if they have high similarity.Assume that the similarity data are represented as a graph G = (V,E) where each vertex represents a species and two vertices are adjacent if they represent species of high similarity. The phylogeny reconstruction problem can then be abstracted as the problem of finding a (phylogenetic) tree T from the given graph G such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) (u, v) ∈ E if and only if dT (u, v) ≤ k for some fixed threshold k, where dT (u, v) denotes the distance between u and v in tree T.This is called the Phylogenetic kth Root Problem (PRk), and such a tree T, if exists, is called a phylogenetic kth root of graph G.The computational complexity of PRk is open, except for k ≤ 4.In this paper, we investigate PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant.Our main contribution is a lineartime algorithm that determines if G has such a phylogenetic kth root, and if so, demonstrates one.On the other hand, as in practice the collected similarity data are usually not perfect and may contain errors, we propose to study a generalized version of PRk where the output phylogeny is only required to be an approximate root of the input graph.W e show that this and other related problems are computationally intractable.