Multiplicative Spanners in Minor-Free Graphs
Greg Bodwin, Gary Hoppenworth, Zihan Tan
TL;DR
The paper studies multiplicative spanners in $K_h$-minor-free graphs and extends known results by establishing a flexible stretch–sparsity/lightness tradeoff. It combines a greedy spanner construction with a density increment theorem for minor-free graphs (building on Postle) to derive tight upper and lower bounds, including an optimal exponent $\frac{2}{k+1}$ under the girth conjecture. The two main contributions are a $(2k-1)$-spanner with sparsity $O\left(h^{\frac{2}{k+1}} \cdot \mathrm{polylog} h\right)$ and a $(1+\varepsilon)(2k-1)$-spanner with lightness $O_\varepsilon\left(h^{\frac{2}{k+1}} \cdot \mathrm{polylog} h\right)$, plus matching lower bounds up to polylog factors. These results advance the understanding of spanners in minor-free graph classes and leverage sophisticated density-increment techniques and hierarchical clustering analyses to achieve near-optimal tradeoffs.
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 $(1+ε)$-spanner of lightness $O_ε(h \sqrt{\log h})$, hence constant when $h$ and $ε$ 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 \[O\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h\right),\] and a $(1+ε)(2k-1)$-spanner with lightness \[O_ε\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h \right).\] 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.