Spanning and Metric Tree Covers Parameterized by Treewidth

TL;DR

The paper advances the theory of tree covers by providing smooth, parameterized tradeoffs between stretch and the number of trees for graphs with small recursive separators or bounded treewidth, including spanning, non-spanning, and HST covers. It leverages separator-based recursive constructions and metric Ramsey theory to achieve near-optimal overlaps, yielding spanning tree covers with stretch as low as O(k log log n) in general graphs and O(k log log t) for treewidth t, with tree counts adapting to the separator size s(n) or t(n). The authors then translate these structural results into practical distance-oracle, path-reporting spanner/emulator, distance-labeling, and routing schemes, achieving improved or competitive stretch/size/query-time tradeoffs for restricted graph families and general graphs. These contributions unify and extend classical results (e.g., TZ01, MN06, ACEFN20) and provide a versatile toolkit for graph metric problems on sparse or structured graphs, with broad applicability to distance queries, label-based querying, and scalable routing. The significance lies in enabling efficient, scalable approximate distance representations and routing in graphs with small separators or bounded treewidth, improving the practicality of spanners, emulators, and labeling schemes in these regimes.

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 $(1+ε)$-approximate non-spanning tree cover with $2^{(t/ε)^{O(t)}}$ 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.

Figures (1)

• Figure 1: The separator $A$ divides the graph $G$ into connected components $C_i$, each containing at most half of the vertices in $G$. Then, the spanning tree cover $\mathcal{T}_A$ from Theorem \ref{['thm:PairwiseSpanTreeCover']} is constructed for the demand set $A$. In addition, a recursive construction of a spanning tree cover $\mathcal{T}_i$ is performed for each component $C_i$. For every two vertices, we consider the tree in $\mathcal{T}_A$ that provides the lowest stretch between them. In case that the two vertices are in the same component $C_i$, we also consider the tree in $\mathcal{T}_i$ that provides the lowest stretch. The weights of these paths, up to stretch $O(k\log\log t(n))$, are the weights of the shortest path that intersects $A$, and the one that does not intersect $A$, respectively. In the drawing, the shortest $a-b$ path intersects $A$, and $a,b$ are not in the same component. Hence, the spanning tree cover $\mathcal{T}_A$ contains a tree that provides stretch $O(k\log\log t(n))$ for $a,b$. For $x,y$, their shortest path does not intersect $A$, and they are in the same component $C_4$. Thus, there is a tree in $\mathcal{T}_4$ that provides stretch $O(k\log\log t(n))$ for $a,b$. Note that the case of $u,v$ might also occur. They are in the same component $C_1$, but their shortest path intersects $A$ (for this reason, the tree that provides low stretch for $u,v$ is in $\mathcal{T}_A$, even though the vertices are in the same component).

Theorems & Definitions (78)

• Definition 1
• Definition 2
• Lemma 1: ES23
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8: Tree Cover
• Definition 9