A Lower Bound for Light Spanners in General Graphs

TL;DR

This paper establishes a lower bound on the lightness of light spanners in general graphs, demonstrating that the dependence on ps cannot be completely eliminated, and highlighting limitations tied to the girth conjecture.

Contribution

It provides the first non-trivial lower bound on lightness that matches the upper bound's dependence on ps, under the girth conjecture, revealing fundamental limitations.

Findings

Lower bound on lightness: b ^{-1/k} n^{1/k} for constant k and specific ps.

Existence of graphs where any 3-spanner has lightness b n^{2/3}.

Limitations on improving bounds without proving the girth conjecture.

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 $\varepsilon:= \Theta(n^{-\frac{1}{2k-1}})$, 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 $\varepsilon$ \emph{or} obtaining an optimal $\varepsilon$-dependence are as hard as proving the girth conjecture for all constant $k$.