A Lower Bound for Light Spanners in General Graphs
Greg Bodwin, Jeremy Flics
TL;DR
This paper establishes that the $\varepsilon$-dependence in lightness bounds for general-graph spanners cannot be removed under the girth conjecture with parameter $k-1$. By constructing a weighted graph from a girth-conjecture graph using a cycle-cluster embedding and random inter-cluster edges, the authors show that for $\varepsilon=\Theta(n^{-1/(2k-1)})$ the lightness of any $(1+\varepsilon)(2k-1)$-spanner is $\Omega\left(\varepsilon^{-1/k} n^{1/k}\right)$, for constant $k\ge2$. A key corollary is that some graphs force $3$-spanners to have lightness $\Omega(n^{2/3})$, improving the previous $\Omega(n^{1/2})$ bound. The analysis hinges on weighted girth, a reduction that ties lightness to graphs of weighted girth $>t+1$, and it reveals that pushing the bound further would imply the girth conjecture for all constant $k$, outlining a barrier to unconditional improvements. Overall, the work clarifies the intrinsic role of $\varepsilon$-dependence in light spanner bounds and introduces a novel embedding-based lower-bound technique under a conditional framework.
Abstract
A recent upper bound by Le and Solomon [STOC '23] has established that every $n$-node graph has a $(1+\varepsilon)(2k-1)$-spanner with lightness $O(\varepsilon^{-1} n^{1/k})$. This bound is optimal up to its dependence on $\varepsilon$; the remaining open problem is whether this dependence can be improved or perhaps even removed entirely. We show that the $\varepsilon$-dependence cannot in fact be completely removed. For constant $k$ and for $\varepsilon:= Θ(n^{-\frac{1}{2k-1}})$, we show a lower bound on lightness of $$Ω\left( \varepsilon^{-1/k} n^{1/k} \right).$$ For example, this implies that there are graphs for which any $3$-spanner has lightness $Ω(n^{2/3})$, improving on the previous lower bound of $Ω(n^{1/2})$. An unusual feature of our lower bound is that it is conditional on the girth conjecture with parameter $k-1$ rather than $k$. We additionally show that this implies certain technical limitations to improving our lower bound further. In particular, under the same conditional, generalizing our lower bound to all $\varepsilon$ \emph{or} obtaining an optimal $\varepsilon$-dependence are as hard as proving the girth conjecture for all constant $k$.