Close the upper–lower bound gap for directed shortcut sets

Close the upper and lower bound gap for directed shortcut sets by improving the current upper or lower bounds for the hopbound achievable with shortcut sets of a given size.

Background

For directed graphs, the paper describes a best-known upper bound of approximately n^{1/3} hopbound for shortcut sets of linear size and a best-known lower bound of approximately n^{1/4}, up to polylogarithmic factors. The discrepancy between these bounds leaves the precise extremal tradeoff unresolved.

The problem is significant because shortcut sets are fundamental structures for reducing hop complexity in directed reachability and shortest-path computations, and determining the correct tradeoff would clarify the limits of existing constructions and lower-bound techniques.