AI 中文总结
研究在最小距离度量下逼近有向图直径问题，提出随机近线性时间算法实现3-逼近，优于此前结果，还扩展到多模图，缩小了最小距离和多模距离逼近差距，为理解非度量距离下的图参数开辟新方向。
AI 中文摘要
我们研究了在最小距离度量（定义为\(d_{\min}(u,v)=\min(d(u,v),d(v,u))\)）下逼近有向图直径的问题。与标准最短路径距离不同，最小距离不是度量，这使得许多经典技术不适用。先前工作致力于逼近此参数，Dalirrooyfard等人[ICALP'19]给出了在\(\tilde{O}(mn^{1/(k + 1)})\)时间内的\(4k - 1\)逼近，Chechik和Zhang[FOCS'22]给出了首个近线性时间常数逼近（4-逼近）。本文提出一种随机近线性时间算法，实现了3-逼近，优于所有已知的逼近-运行时权衡。还将技术扩展到多模图的更一般设置，对于有向2-模图，在近线性时间内获得了3-逼近，显著缩小了最小距离和多模距离逼近之间的差距，为理解非度量距离度量下的图参数开辟了新方向。
英文摘要
We study the problem of approximating the diameter of directed graphs under the min-distance measure, defined as $d_{\min}(u,v) = \min(d(u,v), d(v,u))$. Unlike standard shortest-path distance, min-distance is not a metric, which renders many classical techniques inapplicable. Prior work has therefore focused on approximating this parameter, culminating in an approximation-runtime tradeoff by Dalirrooyfard et al. [ICALP'19] giving a $4k-1$ approximation in $\tilde{O}(mn^{1/(k+1)})$ time for any positive integer $k$ and, more recently, the first near-linear time constant approximation by Chechik and Zhang [FOCS'22], where they obtained a 4-approximation to the min-diameter. In this work we present a randomized near-linear time algorithm that achieves a $3$-approximation to the min-diameter, outperforming all known approximation-runtime tradeoffs. Our approach introduces a novel type-classification framework that may be of independent interest. We further extend our techniques to the more general setting of multimode graphs, recently introduced as a generalization of min-distance by Kirkpatrick and Vassilevska W. [MFCS'25]. For directed $2$-mode graphs, we obtain a $3$-approximation to the diameter in near-linear time, dramatically improving over the previously best known $n$-approximation. Our results significantly narrow the gap between min-distance and multimode distance approximations, and open new directions for understanding graph parameters under non-metric distance measures.
