New Heuristics for Phylogeny Estimation under the Balanced Minimum Evolution Criterion

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 $n\geq3$ must satisfy to encode the \emph{Path-Length Matrix} (PLM) of an \emph{Unrooted Binary Tree} (UBT). 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.