Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC'21] and [Dalirrooyfard and Wein, STOC'21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer $k\ge 2$ and $\delta>0$, a $2-\frac{1}{k}-\delta$ approximation for diameter in directed $m$-edge graphs requires $mn^{1+1/(k-1)-o(1)}$ time. In particular, the simple linear time $2$-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC'13], [Li'20] and [Bonnet, ICALP'21] who proved the first few cases of the curve, $k=2,3$ and $4$, respectively. Our result shows in particular that the simple linear time $2$-approximation algorithm is also optimal for undirected graphs. To obtain our result we develop new tools for fine-grained reductions that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.

中文翻译：

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近似直径的硬度：现在用于无向图

近似图形直径是理论和实践兴趣的基本任务。通过从任意顶点运行 BFS，一个简单的民间传说算法可以在线性时间内输出直径的 2 近似值。在接近线性的时间内是否可能有更好的近似值一直是开放的。一系列关于细粒度复杂性的论文导致了有向图中直径的强硬度结果，最终由 [Li, STOC'21] 和 [Dalirrooyfard and Wein, STOC'21] 独立发现的最近权衡曲线表明，在强指数时间假设 (SETH) 下，对于任何整数 $k\ge 2$ 和 $\delta>0$，$2-\frac{1}{k}-\delta$ 近似于有向 $m$ 中的直径-edge 图需要 $mn^{1+1/(k-1)-o(1)}$ 时间。特别是，简单的线性时间 $2$-approximation 算法是有向图的最佳选择。在本文中，我们证明了同样的权衡下界曲线对于无向图也是可能的，扩展了 [Roditty 和 Vassilevska W., STOC'13]、[Li'20] 和 [Bonnet, ICALP'21] 的结果，他们证明了曲线的前几个案例，分别为 $k=2,3$ 和 $4$。我们的结果特别表明，简单的线性时间 $2$-近似算法对于无向图也是最佳的。为了获得我们的结果，我们开发了用于细粒度缩减的新工具，这些工具可用于证明与距离计算相关的无向图中其他问题的基于 SETH 的硬度。STOC'13]、[Li'20] 和 [Bonnet, ICALP'21] 分别证明了曲线的前几个案例，$k=2,3$ 和 $4$。我们的结果特别表明，简单的线性时间 $2$-近似算法对于无向图也是最佳的。为了获得我们的结果，我们开发了用于细粒度缩减的新工具，这些工具可用于证明与距离计算相关的无向图中其他问题的基于 SETH 的硬度。STOC'13]、[Li'20] 和 [Bonnet, ICALP'21] 分别证明了曲线的前几个案例，$k=2,3$ 和 $4$。我们的结果特别表明，简单的线性时间 $2$-近似算法对于无向图也是最佳的。为了获得我们的结果，我们开发了用于细粒度缩减的新工具，这些工具可用于证明与距离计算相关的无向图中其他问题的基于 SETH 的硬度。