Close the approximation gap for directed weighted replacement paths

Determine the exact randomized round complexity of computing (1+ε)-approximate replacement path distances for a given s–t path in directed weighted graphs in the CONGEST model by closing the current gap between the O~(n^{2/3}+D) upper bound and the Ω~(√n+D) lower bound.

Background

The paper proves a tight Θ~(n^{2/3}+D) bound for exact replacement paths in unweighted directed graphs and gives an O~(n^{2/3}+D) algorithm for (1+ε)-approximate replacement paths in directed weighted graphs.

However, the lower-bound technique used for exact computations does not extend to approximations, leaving a gap between the current upper bound and the best-known Ω~(√n+D) lower bound from prior work.