New Additive Spanner Lower Bounds by an Unlayered Obstacle Product

TL;DR

This paper establishes new lower bounds for additive spanners, showing limitations on how sparse such graphs can be while preserving approximate distances, through novel obstacle product constructions.

Contribution

It introduces a new technique for analyzing obstacle product frameworks, enabling the construction of graphs that improve existing lower bounds for additive spanners.

Findings

Any sparse spanner must increase some distances by at least  n^{1/7}.

The lower bounds hold even with small additive errors for demand pairs.

New analysis methods allow non-layered graphs in obstacle product constructions.

Abstract

For an input graph $G$, an additive spanner is a sparse subgraph $H$ whose shortest paths match those of $G$ up to small additive error. We prove two new lower bounds in the area of additive spanners: 1) We construct $n$-node graphs $G$ for which any spanner on $O(n)$ edges must increase a pairwise distance by $+\Omega(n^{1/7})$. This improves on a recent lower bound of $+\Omega(n^{1/10.5})$ by Lu, Wein, Vassilevska Williams, and Xu [SODA '22]. 2) A classic result by Coppersmith and Elkin [SODA '05] proves that for any $n$-node graph $G$ and set of $p = O(n^{1/2})$ demand pairs, one can exactly preserve all pairwise distances among demand pairs using a spanner on $O(n)$ edges. They also provided a lower bound construction, establishing that that this range $p = O(n^{1/2})$ cannot be improved. We strengthen this lower bound by proving that, for any constant $k$, this range of $p$ is still unimprovable even if the spanner is allowed $+k$ additive error among the demand pairs. This negatively resolves an open question asked by Coppersmith and Elkin [SODA '05] and again by Cygan, Grandoni, and Kavitha [STACS '13] and Abboud and Bodwin [SODA '16]. At a technical level, our lower bounds are obtained by an improvement to the entire obstacle product framework used to compose "inner" and "outer" graphs into lower bound instances. In particular, we develop a new strategy for analysis that allows certain non-layered graphs to be used in the product, and we use this freedom to design better inner and outer graphs that lead to our new lower bounds.