Abstract
The balanced minimal evolution (BME) method of creating phylogenetic trees can be formulated as a linear programming problem, minimizing an inner product over the vertices of the BME polytope. In this paper we undertake the project of describing the facets of this polytope. We classify and identify the combinatorial structure and geometry (facet inequalities) of all the facets in dimensions up to five, and classify even more facets in all dimensions. A full set of facet inequalities would allow a full implementation of the simplex method for finding the BME tree—although there are reasons to think this an unreachable goal. However, our results provide the crucial first steps for a more likely-to-be-successful program: finding efficient relaxations of the BME polytope.
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Acknowledgments
We would like to thank the referees, whose suggestions helped to improve the readability of this paper. The first author would like to thank the organizers and participants in the Working Group for geometric approaches to phylogenetic tree reconstructions, at the NSF/CBMS Conference on Mathematical Phylogeny held at Winthrop University in June–July 2014. Especially helpful were conversations with Ruriko Yoshida, Terrell Hodge and Matt Macauley. The first author would also like to thank the American Mathematical Society and the Mathematical Sciences Program of the National Security Agency for supporting this research through Grant H98230-14-0121. (This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.) The first author’s specific position on the NSA is published in Forcey (2014). Suffice it to say here that he appreciates NSA funding for open research and education, but encourages reformers of the NSA who are working to ensure that protections of civil liberties keep pace with intelligence capabilities.
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Forcey, S., Keefe, L. & Sands, W. Facets of the balanced minimal evolution polytope. J. Math. Biol. 73, 447–468 (2016). https://doi.org/10.1007/s00285-015-0957-1
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DOI: https://doi.org/10.1007/s00285-015-0957-1