# L-Infinity optimization to Bergman fans of matroids with an application to phylogenetics

## TL;DR

This paper explores the geometric structure of ultrametrics as tropical polytopes within Bergman fans of matroids, providing polynomial-time tests for ultrametric tree structures and extending to partial data scenarios.

## Contribution

It introduces a polynomial-time method to verify if all ultrametrics closest to a dissimilarity map share the same tree structure, generalizing to Bergman fans of matroids.

## Key findings

- Provides an internal description of the tropical polytope of ultrametrics.
- Derives a polynomial-time test for ultrametric tree structure consistency.
- Extends results to cases with partial dissimilarity data.

## Abstract

Given a dissimilarity map $\delta$ on finite set $X$, the set of ultrametrics (equidistant tree metrics) which are $l^\infty$-nearest to $\delta$ is a tropical polytope. We give an internal description of this tropical polytope which we use to derive a polynomial-time checkable test for the condition that all ultrametrics $l^\infty$-nearest to $\delta$ have the same tree structure. It was shown by Ardila and Klivans \cite{ardila-klivans2006} that the set of all ultrametrics on a finite set of size $n$ is the Bergman fan associated to the matroid underlying the complete graph on $n$ vertices. Therefore, we derive our results in the more general context of Bergman fans of matroids. This added generality allows our results to be used on dissimilarity maps where only a subset of the entries are known.