SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 701-720, January 2020.
Given a dissimilarity map $\delta$ on a 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 [J. Combin. Theory Ser. B, 96 (2006), pp. 38--49] that the set of all ultrametrics on a finite set of size $n$ is the Bergman fan associated with 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 subsets of the entries are known.

中文翻译：

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拟阵拟贝格扇的L-无穷优化及其在系统发生学中的应用

SIAM离散数学杂志，第34卷，第1期，第701-720页，2020年1月。
给定有限集$ X $上的相异图$ \ delta $，与＃$ delta最接近$ l ^ infty $的超度量（等量树度量）的集合是热带多面体。我们给出了这个热带多面体的内部描述，我们用它来推导多项式时间可检验的测试，条件是所有超度量$ l ^ \ infty $-最近的$ \ delta $具有相同的树结构。它由Ardila和Klivans展示[J. 组合 理论系列 B，96（2006），pp。38--49]，即在大小为$ n $的有限集上的所有超度量集是与在$ n $顶点上的完整图形下方的拟阵有关的Bergman扇形。因此，我们是在更一般的拟人贝格曼迷的一般情况下得出我们的结果。这种增加的通用性使我们的结果可用于仅条目子集已知的不相似图上。