Extreme rays of the $\ell^\infty$-nearest ultrametric tropical polytope

The set of ultrametrics on $[n]$ nodes that are $\ell^\infty$-nearest to a given dissimilarity map forms a $(\max,+)$ tropical polytope. Previous work of Bernstein has given a superset of the set containing all the phylogenetic trees that are extreme rays of this polytope. In this paper, we show that Bernstein's necessary condition of tropical extreme rays is sufficient only for $n=3$ but not for $n\geq 4$. Our proof relies on the exterior description of this tropical polytope, together with the tangent hypergraph techniques for extremality characterization. The sufficiency of the case $n=3$ is proved by explicitly finding all extreme rays through the exterior description. Meanwhile, an inductive construction of counterexamples is given to show the insufficiency for $n\geq 4$.