Subdominant Matroid Ultrametrics

Abstract.

Given a matroid M on the ground set E, the Bergman fan $\mathop \mathcal{B}\limits^ \sim (M),$ or space of M-ultrametrics, is a polyhedral complex in $\mathbb{R}^E$ which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point ω in $\mathbb{R}^E ,$ we wish to find an M-ultrametric which is closest to it in the $\ell _\infty$ -metric. The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A procedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph K _n , the Bergman fan $\mathop \mathcal{B}\limits^ \sim (K_n )$ parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet’s method for computing the tree that most closely matches measured data.