Subsetwise and Multi-Level Additive Spanners with Lightness Guarantees

TL;DR

The paper tackles subsetwise additive spanners with lightness guarantees by introducing Steiner-lightness and subset-lightness, and then developing deterministic and randomized constructions. It delivers three main subsetwise spanner designs: a $+\epsilon W$ spanner with $O_\epsilon(|S|)$ subset-lightness, a $(4+\epsilon)W$ spanner with $O_\epsilon(|V_H|^{1/3}|S|^{1/3})$ subset-lightness, and a sampling-based $(4+\epsilon)W_{\max}$ spanner with $\tilde{O}_\epsilon(|S|\sqrt{|V'_H|/|V_H|})$ subset-lightness, all built on a Steiner-tree based framework and graph scaling. It also introduces an $e$-approximation algorithm for multi-level spanners, improving the prior 4-approximation by leveraging a parameterized rounding and merging strategy. The work clarifies the relationship between Steiner-lightness and subset-lightness and demonstrates that favorable subset-lightness can be achieved in subsetwise settings where Steiner-lightness alone would be inconclusive. Overall, the results enable efficient, lightweight distance-preserving structures for subset of terminals in weighted graphs, with potential applications in network design and visualization.

Abstract

An \emph{additive +$βW$ spanner} of an edge weighted graph $G=(V,E)$ is a subgraph $H$ of $G$ such that for every pair of vertices $u$ and $v$, $d_{H}(u,v) \le d_G(u,v) + βW$, where $d_G(u,v)$ is the shortest path length from $u$ to $v$ in $G$. While additive spanners are very well studied in the literature, spanners that are both additive and lightweight have been introduced more recently [Ahmed et al., WG 2021]. Here the \emph{lightness} is the ratio of the spanner weight to the weight of a minimum spanning tree of $G$. In this paper, we examine the widely known subsetwise setting when the distance conditions need to hold only among the pairs of a given subset $S$. We generalize the concept of lightness to subset-lightness using a Steiner tree and provide polynomial-time algorithms to compute subsetwise additive $+εW$ spanner and $+(4+ε) W$ spanner with $O_ε(|S|)$ and $O_ε(|V_H|^{1/3} |S|^{1/3})$ subset-lightness, respectively, where $ε$ is an arbitrary positive constant. We next examine a multi-level version of spanners that often arises in network visualization and modeling the quality of service requirements in communication networks. The goal here is to compute a nested sequence of spanners with the minimum total edge weight. We provide an $e$-approximation algorithm to compute multi-level spanners assuming that an oracle is given to compute single-level spanners, improving a previously known 4-approximation [Ahmed et al., IWOCA 2023].

Figures (3)

• Figure 1: (a) A graph $G$, where the subset $S$ is shown in squares. A subgraph corresponding to the pairwise shortest paths determined by $S$ is highlighted in blue. Here $W_G(a,b)=W_G(a,d)=W_G(b,d)=10$ and $W_G(a,c)=W_G(b,c)=W_G(c,d)=20$. (b) A $+2W(\cdot,\cdot)$-spanner (in bold) of lightness 50/60$= 0.83$ considering a spanning tree of the blue subgraph as the Steiner tree. (c) Illustration for a multi-level spanner.
• Figure 3: Illustration for (a) Steiner tree $H$, (b) scaled Steiner tree $H_s$, and (c) subdivided Steiner tree $H'$, where the Squares represent vertices of $S$ and solid circles represent subdivision vertices.
• Figure 4: Illustration for the set up for (a) Lemma \ref{['lem:neighbor_improvement']} and (b) Lemma \ref{['lem:neighbor_improvement_6']}. The missing edges are shown in dotted lines.

Theorems & Definitions (20)

• Definition 1: Steiner-Lightness
• Definition 2: Subset-Lightness
• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8