The paper shows fine-grained hardness for approximating reachability diameter in directed graphs, gives additive approximations for unweighted cases, and constant-factor approximations for bounded treewidth and width-bounded DAGs.
Giving some slack: Shortcuts and transitive closure compressions
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Revisiting Diameter in Directed Graphs
The paper shows fine-grained hardness for approximating reachability diameter in directed graphs, gives additive approximations for unweighted cases, and constant-factor approximations for bounded treewidth and width-bounded DAGs.