Constant-factor sublinear approximation for metric Steiner Forest

Improve the approximation ratio for estimating the optimal cost of metric Steiner Forest from the currently known $O(\log k)$ factor to an $O(1)$ factor using sublinear time, thereby avoiding the logarithmic loss arising from scale-based estimators.

Background

The paper presents an $O(\log k)$-approximation algorithm for metric Steiner Forest in near-linear time under the distance-matrix query model. This matches the approximation guarantee of the prior sublinear algorithm cited by the authors.

The logarithmic loss comes from considering $O(\log k)$ distance scales. The authors state that this loss is inherent to the scale-based estimators used in the existing approaches, so achieving an $O(1)$ approximation would require a fundamentally different way to obtain information about the optimal Steiner Forest cost.