Spanning and Metric Tree Covers Parameterized by Treewidth

TL;DR

This paper introduces a new framework for spanning and metric tree covers parameterized by treewidth and separator size, achieving a smooth tradeoff between stretch and the number of trees, with applications to spanners and routing.

Contribution

It presents a unified approach to construct tree covers with adjustable stretch and size based on graph parameters like treewidth and separators, improving previous bounds.

Findings

Achieves tree covers with stretch O(k) and size depending on s(n) or t(n).

Provides spanning tree covers with stretch O(k log log n).

Improves path-reporting spanners, emulators, and routing schemes.

Abstract

Given a graph $G=(V,E)$, a tree cover is a collection of trees $\mathcal{T}=\{T_1,T_2,...,T_q\}$, such that for every pair of vertices $u,v\in V$ there is a tree $T\in\mathcal{T}$ that contains a $u-v$ path with a small stretch. If the trees $T_i$ are sub-graphs of $G$, the tree cover is called a spanning tree cover. If these trees are HSTs, it is called an HST cover. In a seminal work, Mendel and Naor [2006] showed that for any parameter $k=1,2,...$, there exists an HST cover, and a non-spanning tree cover, with stretch $O(k)$ and with $O(kn^{\frac{1}{k}})$ trees. Abraham et al. [2020] devised a spanning version of this result, albeit with stretch $O(k\log\log n)$. For graphs of small treewidth $t$, Gupta et al. [2004] devised an exact spanning tree cover with $O(t\log n)$ trees, and Chang et al. [2-23] devised a $(1+\epsilon)$-approximate non-spanning tree cover with $2^{(t/\epsilon)^{O(t)}}$ trees. We prove a smooth tradeoff between the stretch and the number of trees for graphs with balanced recursive separators of size at most $s(n)$ or treewidth at most $t(n)$. Specifically, for any $k=1,2,...$, we provide tree covers and HST covers with stretch $O(k)$ and $O\left(\frac{k^2\log n}{\log s(n)}\cdot s(n)^{\frac{1}{k}}\right)$ trees or $O(k\log n\cdot t(n)^{\frac{1}{k}})$ trees, respectively. We also devise spanning tree covers with these parameters and stretch $O(k\log\log n)$. In addition devise a spanning tree cover for general graphs with stretch $O(k\log\log n)$ and average overlap $O(n^{\frac{1}{k}})$. We use our tree covers to provide improved path-reporting spanners, emulators (including low-hop emulators, known also as low-hop metric spanners), distance labeling schemes and routing schemes.