Stronger Lower Bounds for Tree Covers via Cyclic Symmetry

Abstract

A tree cover of an $n$-point metric space is a collection of $k$ dominating trees such that every pairwise distance is approximately preserved by at least one tree. The best known general upper bound on the distortion is $\widetilde{O}(n^{1/k})$. Recently, Chen, Tan, and Xu (ITCS 2026, SICOMP 2026) proved a lower bound of $Ω_k(n^{1/2^{k-1}})$ using a topological approach. We improve their lower bound to $Ω_k(n^{1/[k(p-1)]})=Ω_k(n^{1/O(k^2)})$, where $p$ is the smallest prime strictly larger than $k$. Thus, the gap between the known upper and lower exponents is reduced from exponential in $k$ to a factor of $O(k)$. Our key observation is a qualitative difference between the antipodal symmetry underlying the binary labels in the previous approach and the cyclic symmetry used here. In the binary setting, every joint label has a unique antipodal partner, whereas every label in $\mathbb{Z}_p^k$ has many partners that differ from it in every coordinate. This flexibility allows an equivariant Borsuk--Ulam-type theorem in only $k(p-1)$ dimensions to produce two nearby vertices with different labels in all $k$ trees. A cyclic unwinding argument then shows that they are far apart in every tree.