The reachability diameter of a directed graph is the maximum finite shortest-path distance across all pairs of vertices. It has received recent attention under this name, although the underlying quantity has long played a central role in shortcut set and parallel reachability. Notably, Bals et al. establish hardness for every polynomial-factor approximation in nearly-linear time for weighted digraphs. Yet they give $n^{1/2+o(1)}$-approximation in the unweighted regime with nearly-linear running time, and their conditional lower bounds rule out certain constants only, leaving open whether polynomial loss is necessary. We give a randomized algorithm that estimates the reachability diameter of an unweighted digraph within a factor $n^{1/3+o(1)}$, with high probability, in $\widetilde{O}(m+n)$ time. We complement our algorithm with a lower bound conditional on the 3-uniform Hyperclique hypothesis, ruling out an $n^{2/15-\varepsilon}$-approximation in $\widetilde{O}(m+n)$ time for every fixed $\varepsilon\in(0,2/15)$, establishing the necessity of polynomial loss for nearly-linear time algorithms.

翻译：暂无翻译