Optimal trade-off for additive Steiner spanners in hyperbolic space

Characterize the optimal relationship between additive error ε and edge count for Steiner spanners on point sets in d-dimensional hyperbolic space H^d. Determine, for fixed dimension d and ε>0, the minimum possible number of edges required by any ε-additive Steiner spanner for n points in H^d, and develop matching constructions or lower bounds that establish this optimal trade-off.

Background

The paper studies Steiner spanners in Euclidean, spherical, and hyperbolic spaces and presents new near-optimal constructions. In hyperbolic space H^d, the authors obtain an ε-additive Steiner spanner with nearly linear size, leveraging hyperbolic quadtrees and transitive closure spanners, but emphasize that the precise optimal trade-off between additive error and edge count is unresolved.

Prior work by Krauthgamer and Lee and by Park and Vigneron demonstrated existence of additive Steiner spanners in hyperbolic settings, but their bounds involve dimensional and inverse Ackermann factors. The authors’ contribution narrows the gap yet explicitly states that identifying the optimal additive-error/size trade-off remains open.