Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In \emph{directed} graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since $d(u,v)$ may not be the same as $d(v,u)$, there are multiple ways to define the problem, the two most natural being the \emph{(one-way) diameter} ($\max_{(u,v)} d(u,v)$) and the \emph{roundtrip diameter} ($\max_{u,v} d(u,v)+d(v,u)$). In this paper we make progress on the outstanding open question for each of them. -- We design the first algorithm for diameter in sparse directed graphs to achieve $n^{1.5-\varepsilon}$ time with an approximation factor better than $2$. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. -- We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a $1.5$-approximation in subquadratic time would refute the All-Nodes $k$-Cycle hypothesis, and any $(2-\varepsilon)$-approximation would imply a breakthrough algorithm for approximate $\ell_{\infty}$-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH.

翻译：计算图的直径（即最大距离）是一个基础问题，在细粒度复杂度理论中处于核心地位。在无向图中，强指数时间假设（SETH）给出的时间与近似度权衡下界与上界非常接近。然而，在有向图中——其中仅有部分上界适用——仍存在更大的差距。由于d(u,v)可能不等于d(v,u)，该问题有多个定义方式，其中最自然的两种是（单向）直径（max_{(u,v)} d(u,v)）和往返直径（max_{u,v} d(u,v)+d(v,u)）。在本文中，我们针对这两个悬而未决的开放问题取得了进展。——我们设计了首个稀疏有向图直径算法，能够在n^{1.5-ε}时间复杂度内实现优于2的近似比。这一新的上界权衡使得有向图情况更接近无向图情况。值得注意的是，这是首个从快速矩阵乘法中获益的稀疏图直径算法。——我们设计了新的难度归约，将往返直径与有向图直径和无向图直径区分开来。特别地，亚二次时间内的1.5-近似将否定全节点k-环假设，而任何(2-ε)-近似将意味着在近似ℓ∞-最近点对问题上取得突破性算法。值得注意的是，这是首个非基于SETH的直径条件下界。