The reachability diameter ($\mathrm{ReachDiam}$) of a directed graph is the maximum distance over all pairs $u,v$ where $v$ is reachable from $u$. This notion is present in the definition of shortcut sets, and the name was recently coined in that context by Haeupler, Jiang, and Saranurak [SOSA 2026]. While this is a very natural notion of diameter in directed graphs, and especially DAGs, it is so far not computationally explored. Other definitions of diameter in directed graphs are either trivial (infinite) in graphs that are not strongly connected (e.g., the classical definition) or are non-trivial only in highly restrictive graph classes (e.g., Min-Diameter). We initiate the problem of computing the (approximate) reachability diameter from a fine-grained complexity point of view. Under certain fine-grained assumptions, we prove that there is no algorithm in time $\mathcal{O}(n^{ω- \varepsilon}$) that gives any approximation of $\mathrm{ReachDiam}$ in weighted graphs. Similarly, there is no algorithm with better than $2$-approximation for unweighted graphs in this time. To supplement this, we provide algorithmic upper bounds that lead to additive approximation of $\mathrm{ReachDiam}$ for unweighted graphs. Hence, we establish a strong separation between the weighted and unweighted cases, which makes this type of diameter different in nature than other known notions. Considering the hardness in general weighted graphs, we also study special graph classes and get small constant approximations for DAGs with bounded width or graphs with bounded treewidth. Interestingly, our techniques also lead to exact hopsets with hopbound $2$ for bounded treewidth graphs. This and some of our upper bounds for general graphs show technical connections between approximating $\mathrm{ReachDiam}$ and computing shortcut sets and hopsets.

翻译：有向图的可达直径（$\mathrm{ReachDiam}$）定义为所有满足$v$可从$u$到达的节点对$u,v$之间的最大距离。这一概念出现在捷径集的定义中，该名称由Haeupler、Jiang和Saranurak [SOSA 2026]近期在其相关研究中提出。尽管这是有向图（尤其是DAG）中一个非常自然的直径概念，但迄今尚未得到计算层面的深入探索。有向图中其他直径定义要么在非强连通图中是平凡的（无穷大，如经典定义），要么仅在高度受限的图类中非平凡（如最小直径）。我们从细粒度复杂度视角出发，首次提出计算（近似）可达直径的问题。在特定细粒度假设下，我们证明：在加权图中，不存在时间复杂度为$\mathcal{O}(n^{ω- \varepsilon})$的算法能对$\mathrm{ReachDiam}$进行任何近似；类似地，在非加权图中，亦无算法能在该时间内实现优于$2$倍的近似。作为补充，我们给出非加权图中可达直径加法近似的算法上界，从而建立了加权图与非加权图之间的显著区分，这使得该直径类型在本质上区别于其他已知概念。考虑到一般加权图的计算难度，我们还研究了特殊图类，并针对有界宽度DAG或有界树宽图获得了小常数近似。有趣的是，我们的技术还为有界树宽图导出了跃界为$2$的精确跳图。这一结果以及我们针对一般图的部分上界，揭示了近似$\mathrm{ReachDiam}$与计算捷径集及跳图之间的技术关联。