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Subdominant matroid ultrametrics Federico Ardila

Summary: Subdominant matroid ultrametrics
Federico Ardila
Given a matroid M on the ground set E, the Bergman fan B(M), or
space of M-ultrametrics, is a polyhedral complex in RE
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 in RE
, we wish to
find an M-ultrametric which is closest to it in the -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 . 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 com-
plete graph Kn, the Bergman fan B(Kn) parameterizes the equidistant
phylogenetic trees with n leaves. In this case, our results provide a con-
ceptual explanation for Chepoi and Fichet's method for computing the


Source: Ardila, Federico - Department of Mathematics, San Francisco State University


Collections: Mathematics