Optimality of the Neighbor Joining Algorithm and Faces of the Balanced Minimum Evolution Polytope

TL;DR

This paper investigates the geometric structure of the BME polytope, revealing how the Neighbor Joining algorithm relates to BME optimality and identifying specific faces and edges that correspond to subtree moves and clades.

Contribution

It proves that subtree-prune-regraft moves correspond to edges of the BME polytope, describes clade-faces as smaller BME polytopes, and establishes the connection between NJ and BME trees.

Findings

Subtree-prune-regraft moves are edges of the BME polytope.

Clade-faces form smaller BME polytopes.

For any tree order, a distance matrix exists where NJ returns the BME tree.

Abstract

Balanced minimum evolution (BME) is a statistically consistent distance-based method to reconstruct a phylogenetic tree from an alignment of molecular data. In 2000, Pauplin showed that the BME method is equivalent to optimizing a linear functional over the BME polytope, the convex hull of the BME vectors obtained from Pauplin's formula applied to all binary trees. The BME method is related to the Neighbor Joining (NJ) algorithm, now known to be a greedy optimization of the BME principle. Further, the NJ and BME algorithms have been studied previously to understand when the NJ Algorithm returns a BME tree for small numbers of taxa. In this paper we aim to elucidate the structure of the BME polytope and strengthen knowledge of the connection between the BME method and NJ Algorithm. We first prove that any subtree-prune-regraft move from a binary tree to another binary tree corresponds to an edge of the BME polytope. Moreover, we describe an entire family of faces parametrized by disjoint clades. We show that these {\em clade-faces} are smaller dimensional BME polytopes themselves. Finally, we show that for any order of joining nodes to form a tree, there exists an associated distance matrix (i.e., dissimilarity map) for which the NJ Algorithm returns the BME tree. More strongly, we show that the BME cone and every NJ cone associated to a tree $T$ have an intersection of positive measure.