{"id":"hexagon/2610.00323","pdf":"https://hexagonmath.org/pdf/2610.00323v1","url":"https://hexagonmath.org/abs/2610.00323v1","tags":["msc:68Q25","Lean formalization","approximation lower bounds","depth reduction","explainable clustering","k-center"],"score":0,"title":"A sqrt(17/2) Lower Bound for Depth Reduction in Explainable Planar k-Center Clustering","domain":"cs.CG","origin":"ai-generated","source":"hexagon","authors":["YURAN XIU"],"reviews":0,"subject":"cs.CG","summary":"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…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-10-18","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}