Subsetwise and Multi-Level Additive Spanners with Lightness Guarantees

2024-11-12

arXiv.org

AI Summary

This paper studies lightweight constructions of subset additive spanners and multi-level spanners in edge-weighted graphs. For a subset $S subseteq V$, we introduce the notion of “subset lightness” based on Steiner trees—its first formal definition—and present polynomial-time algorithms that construct subset spanners with additive stretch $+varepsilon W$ and $+(4+varepsilon)W$, achieving lightness $O_varepsilon(|S|)$ and $O_varepsilon(|V_H|^{1/3}|S|^{1/3})$, respectively. For multi-level spanners, we design the first $e$-approximation algorithm—improving upon the previous best approximation ratio of 4—yielding a solution whose total edge weight is at most $e$ times the optimum. All algorithms are provably correct, run in polynomial time, and provide rigorous theoretical guarantees on both stretch and lightness.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

An emph{additive +$eta W$ spanner} of an edge weighted graph $G=(V,E)$ is a subgraph $H$ of $G$ such that for every pair of vertices $u$ and $v$, $d_{H}(u,v) le d_G(u,v) + eta W$, 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 $+epsilon W$ spanner and $+(4+epsilon) W$ spanner with $O_epsilon(|S|)$ and $O_epsilon(|V_H|^{1/3} |S|^{1/3})$ subset-lightness, respectively, where $epsilon$ 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].

Problem

• Develop subsetwise additive spanners
• Generalize lightness using Steiner trees
• Provide e-approximation for multi-level spanners

Innovation

• Subsetwise additive spanners
• Multi-level spanners
• Polynomial-time algorithms