New heuristics for phylogeny estimation under the balanced minimum evolution criterion

Abstract

Recent advances in the combinatorics of the Balanced Minimum Evolution Problem (BMEP) enabled the characterization of the mathematical properties that a symmetric integer matrix of order n≥3 must satisfy to encode the Path-Length Matrix of an Unrooted Binary Tree. This result, together with the identification of fundamental facet-defining inequalities for the convex hull of BMEP solutions, has led to an integer linear programming formulation that currently serves as the reference exact solution algorithm. Here, we show how to exploit these advances to improve the approximation algorithms for the problem. We first leverage the tight linear programming relaxation of this formulation to develop an enhanced Neighbor Joining-like heuristic. Next, we embed this heuristic into a Beam Search framework to further improve the quality of the solutions. Computational experiments show that the proposed algorithms outperform existing heuristics, making their use highly desirable in practice.

Availability and implementation: Codes and data are available at https://github.com/HenriDeh/BME_BeamSearch.git and archived at https://zenodo.org/records/15631441 (DOI: 10.5281/zenodo.15631440).

Figure 1.

An example of a phylogeny of five taxa represented by a UBT with 5 leaves and the associated PLM $\mathit{\tau }$.

Figure 2.

The three iterations of an agglomeration method required to generate a phylogeny of six taxa: iteration (1) merges sigletons $\left\{1\right\}$ and $\left\{2\right\}$ and introduces internal node A; iteration (2) merges cluster $\{1,2\}$ with the singleton $\left\{3\right\}$ and introduces node B; iteration (3) merges singletons $\left\{5\right\}$ and $\left\{6\right\}$ and introduces node C; and (4) resulting in the final 6-taxon phylogeny.