Close the complexity gap for (1+ε)-approximate replacement paths in weighted directed graphs

Determine the exact randomized round complexity of the (1+ε)-approximate Replacement Paths problem in the CONGEST model on weighted directed graphs by closing the current gap between the upper bound \widetilde{O}(n^{2/3} + D) and the lower bound \widetilde{\Omega}(\sqrt{n} + D).

Background

The paper establishes tight bounds of \widetilde{\Theta}(n^{2/3}+D) for exact replacement paths in unweighted directed graphs and gives an \widetilde{O}(n^{2/3}+D) upper bound for (1+ε)-approximate replacement paths in weighted directed graphs.

However, their lower bound technique does not extend to approximation settings, while prior work provides a \widetilde{\Omega}(\sqrt{n}+D) lower bound. This leaves a nontrivial gap for the weighted directed approximation case that needs to be closed to identify the exact complexity.