Additive Approximation for Near-Perfect Phylogeny Construction

TL;DR

This paper presents a polynomial-time algorithm that constructs near-optimal phylogenetic trees with additive approximation guarantees, broadening efficient solutions for datasets with near-perfect phylogeny.

Contribution

It introduces the first polynomial-time algorithm achieving an additive approximation of $d+O(q^2)$ for near-perfect phylogeny construction, improving prior exponential-time methods.

Findings

Provides a polynomial-time $(1+o(1))$-approximation algorithm

Broadens the range of datasets with efficient near-optimal solutions

Achieves additive approximation guarantees for near-perfect phylogenies

Abstract

We study the problem of constructing phylogenetic trees for a given set of species. The problem is formulated as that of finding a minimum Steiner tree on $n$ points over the Boolean hypercube of dimension $d$. It is known that an optimal tree can be found in linear time if the given dataset has a perfect phylogeny, i.e. cost of the optimal phylogeny is exactly $d$. Moreover, if the data has a near-perfect phylogeny, i.e. the cost of the optimal Steiner tree is $d+q$, it is known that an exact solution can be found in running time which is polynomial in the number of species and $d$, yet exponential in $q$. In this work, we give a polynomial-time algorithm (in both $d$ and $q$) that finds a phylogenetic tree of cost $d+O(q^2)$. This provides the best guarantees known - namely, a $(1+o(1))$-approximation - for the case $\log(d) \ll q \ll \sqrt{d}$, broadening the range of settings for which near-optimal solutions can be efficiently found. We also discuss the motivation and reasoning for studying such additive approximations.