Dice Question Streamline Icon: https://streamlinehq.com

Beat the O~(n^{2/3}+D) bound for the reachability version of replacement paths

Develop a distributed algorithm that beats the O~(n^{2/3}+D) randomized round complexity for the reachability version of the replacement path problem in the CONGEST model, or establish matching lower bounds that show such improvement is impossible.

Information Square Streamline Icon: https://streamlinehq.com

Background

Beyond exact distances, the authors consider approximation and even the simpler reachability variant (existence of an s–t path avoiding a given edge).

They explicitly state that even for this reachability variant, no approach is known to surpass the O~(n{2/3}+D) upper bound.

References

Is it possible to break the upper bound \widetilde{O}(n{2/3} + D) for any constant-factor approximation? Even for the reachability version of the replacement path problem, we do not know how to break the upper bound \widetilde{O}(n{2/3} + D).

— Optimal Distributed Replacement Paths (2502.15378 - Chang et al., 21 Feb 2025) in Section: Conclusions and Open Problems (Approximation algorithms)