Geometric spanners of bounded tree-width

TL;DR

This work studies the design of geometric spanners whose underlying graphs have bounded tree-width, aiming for small dilation while enabling efficient algorithms. The authors introduce a trade-off framework in which, for fixed dimension $d\ge 2$ and target tree-width $k$, one can construct a spanner of dilation $O\left(n/k^{d/(d-1)}\right)$ in time $O(n^2\log n)$, and prove a matching lower bound that this dependence is asymptotically optimal. They extend the construction to planar spanners with bounded degree and dilation $O(n/k^2)$ via minor-3-core techniques, and derive sharp results for convex and circular point sets, including a tree-spanner on circle points with dilation near $2n/\pi$. The paper also develops structural tools (minor-3-cores) to bound the tree-width of planar spanners and demonstrates strong implications for bounded-tree-width geometric problems, with several open questions on optimization and approximation. Overall, it advances the tractable design of geometric spanners under tree-width constraints and clarifies the fundamental dilation–tree-width trade-offs in Euclidean spaces.

Abstract

Given a point set $P$ in the Euclidean space, a geometric $t$-spanner $G$ is a graph on $P$ such that for every pair of points, the shortest path in $G$ between those points is at most a factor $t$ longer than the Euclidean distance between those points. The value $t\geq 1$ is called the dilation of $G$. Commonly, the aim is to construct a $t$-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation $t$. Let $d$ be a fixed integer and $P \subset \mathbb{R}^d$ be a point set with $n$ points. We give a first algorithm to compute an $\mathcal{O}(n/k^{d/(d-1)})$-spanner on $P$ with tree-width at most $k$. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width $k$: We show that there is a set of $n$ points such that every spanner of tree-width $k$ has dilation $\mathcal{O}(n/k^{d/(d-1)})$. We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in $\mathbb{R}^2$, a plane spanner with tree-width at most $k$ and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of $n$ equally spaced points on a circle, answering an open question asked in previous work.

Figures (3)

• Figure 1: The set of points resembling the $(4+1)^2$-grid where $n=240$ and $m=6$.
• Figure 2: Grid points $p$ and $q$ as well as the three hypercubes centered at $p$, $q$ and $s$. The subpath of $\gamma_{p,q}$ that is contracted to $p$ is shown in green and the subpath that is contracted to $q$ in purple. The red path is a detour of length $\leq m/4$ for points $p_i$ and $p_{i+1}$.
• Figure 3: Minor-3-core of a graph $G$ that consists of a tree with 4 additional edges (orange).

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 3
• Lemma 4: Dvok2019
• Lemma 5: Le2024
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9