Spanning and Metric Tree Covers Parameterized by Treewidth

Given a graph $G=(V,E)$, a tree cover is a collection of trees $\mathcal{T}=\{T_1,T_2,\mathrm{.}\mathrm{.}\mathrm{.},T_q\}$, 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 $k=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, there exists an HST cover, and a non-spanning tree cover, with stretch $O(k)$ and with $O(k{n}^{\frac{1}{k}})$ 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 $2^{(t\mathrm{/}\epsilon {)}^{O(t)}}$ 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 $k=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, we provide tree covers and HST covers with stretch $O(k)$ and $O(\frac{k^2\log n}{\log s(n)}\cdot s(n{)}^{\frac{1}{k}})$ 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.