A sqrt(17/2) Lower Bound for Depth Reduction in Explainable Planar k-Center Clustering
We prove that the factor-$2$ lower bound for depth reduction in explainable planar $k$-center clustering is not tight. For every integer $k\ge2$, an explicit set of $4k$ distinct integer points has both ordinary and explainable $k$-center radius one, while every axis-parallel threshold tree with at most $k$ nonempty leaves and depth less than $k-1$ has radius at least $\sqrt{17/2}$. Thus, for every fixed depth multiplier $H$ and every $C<\sqrt{17/2}$, arbitrarily large instances admit no such tree of depth at most $H\lceil\log_2(k+1)\rceil$ and radius at most $C$ times the optimal…
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