Simple Reconstruction of Binary Near-Perfect Phylogenetic Trees

Abstract

Abstract.We consider the problem of reconstructing near-perfect phylogenetic trees using binary character states (referred to as BNPP). A perfect phylogeny assumes that every character mutates at most once in the evolutionary tree, yielding an algorithm for binary character states that is computationally efficient but not robust to imperfections in real data. A near-perfect phylogeny relaxes the perfect phylogeny assumption by allowing at most a constant number q of additional mutations. In this paper, we develop an algorithm for constructing optimal phylogenies and provide empirical evidence of its performance. The algorithm runs in time O((72κ) q nm + nm 2 ) where n is the number of taxa, m is the number of characters and κ is the number of characters that share four gametes with some other character. This is fixed parameter tractable when q and κ are constants and significantly improves on the previous asymptotic bounds by reducing the exponent to q. Furthermore, the complexity of the previous work makes it impractical and in fact no known implementation of it exists. We implement our algorithm and demonstrate it on a selection of real data sets, showing that it substantially outperforms its worstcase bounds and yields far superior results to a commonly used heuristic method in at least one case. Our results therefore describe the first practical phylogenetic tree reconstruction algorithm that finds guaranteed optimal solutions while being easily implemented and computationally feasible for data sets of biologically meaningful size and complexity.

Mixed Integer Linear Programming for Maximum-Parsimony Phylogeny Inference

“…Even in this restricted setting, the Steiner tree problem has been shown to be NP-complete [21]. However, the phylogeny reconstruction problem when the optimal phylogeny is q-near-perfect can be solved in time polynomial in n and m when q = O(log(poly(n, m))) [17]. If q is very large, though, such algorithms do not perform well.…”

“…The standard in practice is the use of sophisticated heuristics that will always produce a tree but cannot guarantee optimality (e.g., [11], [12], [13]). Some theoretical advances have recently been made in the efficient solution of near-perfect phylogenies, those that deviate only by a fixed amount from the assumption of perfection [14], [15], [16], [17]. These methods can provide provably efficient solutions in many instances, but still struggle with some moderate-size data sets in practice.…”

Efficiently Finding the Most Parsimonious Phylogenetic Tree Via Linear Programming

“…Rather, the major determinant of run time appears to be a dataset's imperfection, i.e., the difference between the optimal length and the number of variant sites. It has recently been shown that the phylogeny problem under various assumptions is fixed parameter tractable in imperfection [6,13,31,32] possibly suggesting why it is a critical factor in run time determination. The pars program of phylip, despite providing no guarantees of optimality, does indeed find optimal phylogenies in all of the above instances.…”

“…Even in this restricted setting, the Steiner tree problem has been shown to be NP-complete [14]. However, the phylogeny reconstruction problem when the optimal phylogeny is q-near-perfect can be solved in time polynomial in n and m when q = O(log(poly(n, m))) [32]. If q is very large, though, such algorithms do not perform well.…”

“…[3,12,27]). Some theoretical advances have recently been made in the efficient solution of near-perfect phylogenies, those that deviate only by a fixed amount from the assumption of perfection [6,13,31,32]. These methods can provide provably efficient solutions in many instances, but still struggle with some moderate-size data sets in practice.…”