A Unified Framework of Light Spanners II: Fine-Grained Optimality

TL;DR

This paper introduces a unified framework for constructing light spanners across various graph classes, improving optimality bounds and runtime efficiency by transforming sparse spanners into light spanners, with significant results for minor-free graphs.

Contribution

It develops a refined, unified approach to light spanners that enhances bounds and efficiency, addressing previous ad hoc methods and sub-optimal runtimes.

Findings

Improved lightness bound for $K_r$-minor-free graphs: $ ilde{O}_{r,rac{1}{ ext{epsilon}}}(rac{r}{ ext{epsilon}} + rac{1}{ ext{epsilon}^2})$

Established a lower bound matching the upper bound for lightness, $ ext{Omega}(rac{r}{ ext{epsilon}} + rac{1}{ ext{epsilon}^2})$

Surprising quadratic dependency on $1/ ext{epsilon}$ in lightness bounds

Abstract

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. Our framework is developed in two papers. The current paper is the second of the two -- it builds on the basis of the unified framework laid in the first paper, and then strengthens it to achieve more refined optimality bounds for several graph classes. Among various applications and implications of our framework, we highlight here the following: For $K_r$-minor-free graphs, we provide a $(1+\epsilon)$-spanner with lightness $\tilde{O}_{r,\epsilon}( \frac{r}{\epsilon} + \frac{1}{\epsilon^2})$, improving the lightness bound $\tilde{O}_{r,\epsilon}( \frac{r}{\epsilon^3})$ of Borradaile, Le and Wulff-Nilsen. We complement our upper bound with a lower bound construction, for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\frac{r}{\epsilon} + \frac{1}{\epsilon^2})$. We note that the quadratic dependency on $1/\epsilon$ we proved here is surprising, as the prior work suggested that the dependency on $\epsilon$ should be $1/\epsilon$.