Lower Bounds on Tree Covers

2025-08-14

AI Summary

This paper investigates the minimum distortion lower bound for covering a metric space with $k$ trees. Prior work established only an $Omega(log_k n)$ lower bound and an $ ilde{O}(n^{1/k})$ upper bound for constant $k geq 2$, leaving a substantial gap. To close this gap, we construct a family of structured grid-like graphs and combine combinatorial fixed-point arguments with refined graph-theoretic analysis. Our main contribution is the first lower bound of $Omega(n^{1/2^{k-1}})$ on distortion, exhibiting doubly exponential decay in $k$. This result significantly tightens the long-standing theoretical gap and provides the strongest known lower bound on the trade-off between distortion and the number of trees in tree coverings. Moreover, it introduces novel analytical tools—bridging combinatorial topology and metric graph theory—that advance the study of metric embeddings and hierarchical metric decompositions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Given an $n$-point metric space $(X,d_X)$, a tree cover $mathcal{T}$ is a set of $|mathcal{T}|=k$ trees on $X$ such that every pair of vertices in $X$ has a low-distortion path in one of the trees in $mathcal{T}$. Tree covers have been playing a crucial role in graph algorithms for decades, and the research focus is the construction of tree covers with small size $k$ and distortion. When $k=1$, the best distortion is known to be $Θ(n)$. For a constant $kge 2$, the best distortion upper bound is $ ilde O(n^{frac 1 k})$ and the strongest lower bound is $Ω(log_k n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $Ω(n^{frac{1}{2^{k-1}}})$. Our proof is a novel analysis on a structurally simple grid-like graph, which utilizes some combinatorial fixed-point theorems. We believe that they will prove useful for analyzing other tree-like data structures as well.

Problem

• Improving lower bounds on tree cover distortion
• Analyzing grid-like graphs for tree covers
• Bridging gap between upper and lower bounds

Innovation

• Novel grid-like graph analysis
• Combinatorial fixed-point theorems
• Improved lower bound proof