Additive Spanner Lower Bounds with Optimal Inner Graph Structure

TL;DR

This paper presents improved lower bounds for additive spanners and emulators, demonstrating that any such sparse structures must incur significant additive errors, thus advancing understanding of their limitations.

Contribution

It introduces new graph constructions with optimal inner graph structures that establish tighter lower bounds for additive spanners and emulators.

Findings

Lower bound for additive spanners improved to +\Omega(n^{3/17})

Lower bound for additive emulators improved to +\Omega(n^{1/14})

Techniques enable matching properties of inner graphs in lower and upper bound frameworks

Abstract

We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for $O(n)$-size additive emulators to $+\Omega(n^{1/14})$.