Remove the log(1/ε) factor in the greedy spanner’s existential lightness optimality

Determine whether the existential lightness optimality ratio of the greedy (1+ε)-spanner in ℝ^d can be improved by eliminating the multiplicative log(1/ε) factor, i.e., establish an upper bound showing that the greedy (1+ε)-spanner’s total weight is O_d(ε^{-d}) times the weight of an MST (up to dimension-dependent constants), rather than O_d(ε^{-d}·log(1/ε)).

Background

The paper reviews existential optimality results for Euclidean spanners, noting that the greedy (1+ε)-spanner achieves the best-known lightness upper bound of O_d(ε^{-d}·log(1/ε)) times the weight of a minimum spanning tree. Known lower bounds show that any (1+ε)-spanner must have lightness at least Ω_d(ε^{-d}), leaving a gap of a log(1/ε) factor between upper and lower bounds.

The authors point out that the greedy spanner is existentially optimal for lightness only up to an extra log(1/ε) factor, and explicitly state that removing this logarithmic term is an open problem. Resolving this would tighten the existential optimality of the greedy spanner for lightness to match the lower bound up to constants depending on dimension.