New Additive Spanner Lower Bounds by an Unlayered Obstacle Product

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.