Multiplicative Spanners in Minor-Free Graphs

2025-04-23

AI Summary

This paper investigates the sparsity–lightness tradeoff for multiplicative spanners on $K_h$-minor-free graphs. For any integer $k geq 1$, we present new constructions of $(2k-1)$-spanners and $(1+varepsilon)(2k-1)$-spanners, leveraging an improved density-increment theorem (building on Postle, 2020) combined with hierarchical clustering and inductive contraction techniques. We establish, for the first time, the optimal $h$-dependence $h^{2/(k+1)} cdot mathrm{polylog},h$ for both sparsity and lightness—tight under the girth conjecture. Our $(2k-1)$-spanner achieves optimal sparsity, while the $(1+varepsilon)(2k-1)$-spanner simultaneously attains matching lightness bounds, significantly improving upon all prior results.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+epsilon)$-spanner of lightness $O_{epsilon}(h sqrt{log h})$, hence constant when $h$ and $epsilon$ are regarded as constants. We extend this result by showing that a more expressive size/stretch tradeoff is available. Specifically: for any positive integer $k$, every $n$-node, $K_h$-minor-free graph has a $(2k-1)$-spanner with sparsity [Oleft(h^{frac{2}{k+1}} cdot ext{polylog } h ight),] and a $(1+epsilon)(2k-1)$-spanner with lightness [O_{epsilon}left(h^{frac{2}{k+1}} cdot ext{polylog } h ight).] We further prove that this exponent $frac{2}{k+1}$ is best possible, assuming the girth conjecture. At a technical level, our proofs leverage the recent improvements by Postle (2020) to the remarkable density increment theorem for minor-free graphs.

Problem

• Extends light spanner results for minor-free graphs
• Proves optimal size/stretch tradeoffs for spanners
• Leverages density increment theorem improvements

Innovation

• Extends light spanners to minor-free graphs
• Introduces expressive size/stretch tradeoff
• Leverages Postle's density increment theorem