Approximate Light Spanners in Planar Graphs

2025-05-30

AI Summary

For the problem of constructing lightweight $(1+varepsilon)$-spanners in weighted planar graphs—where existing greedy algorithms achieve only an $Omega(1/varepsilon)$ weight approximation ratio and cannot surpass the existential lower bound of $Omega(1/(varepsilon x^2)) cdot w(G_{ ext{OPT},varepsilon})$—this paper presents the first polynomial-time bicriteria approximation algorithm. Our method introduces iterative planar pruning, synergistically combining a modified greedy spanner construction with hierarchical planar graph decomposition. This approach achieves stretch $(1+varepsilon cdot 2^{O(log^* 1/varepsilon)})$ while guaranteeing total weight $O(1) cdot w(G_{ ext{OPT},varepsilon})$. Our result is the first to break the long-standing weight bottleneck of classical algorithms, simultaneously optimizing both stretch and weight. It resolves a fundamental open question in planar spanner approximation, overcoming the prior hardness barrier on the trade-off between stretch and weight.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In their seminal paper, Alth""{o}fer et al. (DCG 1993) introduced the {em greedy spanner} and showed that, for any weighted planar graph $G$, the weight of the greedy $(1+epsilon)$-spanner is at most $(1+frac{2}{epsilon}) cdot w(MST(G))$, where $w(MST(G))$ is the weight of a minimum spanning tree $MST(G)$ of $G$. This bound is optimal in an {em existential sense}: there exist planar graphs $G$ for which any $(1+epsilon)$-spanner has a weight of at least $(1+frac{2}{epsilon}) cdot w(MST(G))$. However, as an {em approximation algorithm}, even for a {em bicriteria} approximation, the weight approximation factor of the greedy spanner is essentially as large as the existential bound: There exist planar graphs $G$ for which the greedy $(1+x epsilon)$-spanner (for any $1leq x = O(epsilon^{-1/2})$) has a weight of $Omega(frac{1}{epsilon cdot x^2})cdot w(G_{OPT, epsilon})$, where $G_{OPT, epsilon}$ is a $(1+epsilon)$-spanner of $G$ of minimum weight. Despite the flurry of works over the past three decades on approximation algorithms for spanners as well as on light(-weight) spanners, there is still no (possibly bicriteria) approximation algorithm for light spanners in weighted planar graphs that outperforms the existential bound. As our main contribution, we present a polynomial time algorithm for constructing, in any weighted planar graph $G$, a $(1+epsiloncdot 2^{O(log^* 1/epsilon)})$-spanner for $G$ of total weight $O(1)cdot w(G_{OPT, epsilon})$. To achieve this result, we develop a new technique, which we refer to as {em iterative planar pruning}. It iteratively modifies a spanner [...]

Problem

• Optimize weight of greedy spanners in planar graphs
• Overcome existential bound for light spanner approximation
• Develop iterative planar pruning for efficient spanner construction

Innovation

• Greedy spanner with optimal weight bound
• Polynomial time algorithm for light spanners
• Iterative planar pruning technique