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.
翻译:图形直径是理论和实际兴趣的一项基本任务。 简单的民俗算法可以通过任意的顶点运行 BFS,在线性时间中输出直径的2个方略值。 在近线性时间里, 是否可能有一个更好的近似值。 一系列精度复杂程度的论文导致定向图形直径的强烈硬性结果, 最终形成一个由[ Li, STOC'21] 和 [ Dalirrooyfard and Wein, STOC'21] 独立发现的最新折价曲线。 表明, 在“ 强烈的指数时间节奏” 下, 可以在直线性时间里输出直径2k\ge 2美元 和 $\\delta>0 。 在本文中, 2\\ frac\ frac{1\ k}\\ delta$ 直径的直径直径, 直径直图需要 $mnn_ +1/1/ (k-1) (k-1)- o) 时间。 。 简单线性时间 $ 2$- approfrofrofrofalfard dalfall dal dal dal dalation 算算法是用于直图的正数。 4, 我们的中, lexalalaldalentalalalalalalalalalalalalaldaldaldaldalationaldalationalationalationalationalationalationalationalationalational 。 。 。 其结果证明 。