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 $\varepsilon:= \Theta(n^{-\frac{1}{2k-1}})$, we show a lower bound on lightness of $$\Omega\left( \varepsilon^{-1/k} n^{1/k} \right).$$ For example, this implies that there are graphs for which any $3$-spanner has lightness $\Omega(n^{2/3})$, improving on the previous lower bound of $\Omega(n^{1/2})$. 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 $\varepsilon$ or obtaining an optimal $\varepsilon$-dependence are as hard as proving the girth conjecture for all constant $k$.

翻译：Le与Solomon [STOC '23] 近期提出的上界表明，每个包含$n$个节点的图都存在一个$(1+\varepsilon)(2k-1)$-生成子图，其轻量级为$O(\varepsilon^{-1} n^{1/k})$。该上界在除$\varepsilon$依赖项外的部分已达到最优；当前悬而未决的问题是这种$\varepsilon$依赖性能否被改进甚至完全消除。我们证明$\varepsilon$依赖性实际上无法被完全消除。对于常数$k$与$\varepsilon:= \Theta(n^{-\frac{1}{2k-1}})$，我们给出了轻量级的下界：$$\Omega\left( \varepsilon^{-1/k} n^{1/k} \right).$$ 例如，这意味着存在某些图，其任意$3$-生成子图的轻量级均为$\Omega(n^{2/3})$，这改进了先前$\Omega(n^{1/2})$的下界。我们所得下界的一个特殊性质在于：其证明条件依赖于参数为$k-1$（而非$k$）的围长猜想。我们进一步证明，这一特性意味着在改进下界方面存在特定的技术限制。具体而言，在相同条件下，将我们的下界推广至所有$\varepsilon$或获得最优的$\varepsilon$依赖性，其难度等同于证明针对所有常数$k$的围长猜想。