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arXiv · 1702.05141

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

Daniel Irving Bernstein
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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.

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Daniel Irving Bernstein. 2017-02-16. L-Infinity optimization to Bergman fans of matroids with an application to phylogenetics. https://arxiv.org/abs/1702.05141

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