Computational Geometry

cs.CG — Computational Geometry

A sqrt(17/2) Lower Bound for Depth Reduction in Explainable Planar k-Center Clustering

Contributed by YURAN XIU

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 explainable radius. The point set is a rescaled tail of the ladder of Deng, Gavva, Karthik, Patel, and Srinivasan. The strengthened analysis uses a four-color packing bound and the exact cluster budget to prove rigidity of every sufficiently small-diameter partition, without an explainability assumption. The complete infinite-family theorem and its unbounded-size corollary have been checked in Lean. The existence of a larger universal constant for logarithmic-depth explanations remains unresolved. HAL deposit: hal-05800242v1.