Additive Spanner Lower Bounds with Optimal Inner Graph Structure
Greg Bodwin, Gary Hoppenworth, Virginia Vassilevska Williams, Nicole Wein, Zixuan Xu
TL;DR
The paper proves a new lower bound for additive $O(n)$-edge spanners, showing there exist graphs on $n$ vertices where any such spanner incurs additive distortion at least $Ω(n^{3/17})$, improving the prior $Ω(n^{1/7})$ bound. It achieves this by a refined obstacle-product construction that combines an unlayered DP LB outer graph with an optimal implicit alternation product and inner graphs based on subset DP LB, aligning the outer- and inner-graph properties with upper-bound techniques. The authors also strengthen emulator lower bounds to $Ω(n^{1/14})$ and clarify how subset-distance-preserver constructions interact with the outer graph, establishing a path toward tighter overall bounds. The results advance the understanding of sparse additive spanners and emulators and outline concrete next steps for fully tight upper and lower bounds via the remaining alignment challenges.
Abstract
We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+Ω(n^{3/17})$, improving on the previous best lower bound of $Ω(n^{1/7})$ [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 $+Ω(n^{1/14})$.