发表机构
MIT（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有向图上的两种相遇距离（最小最大距离和最小总距离）下的直径、半径和离心率，给出一般有向图的最优2近似算法，有向无环图的线性时间精确或近似算法，以及k点相遇直径的通用近似算法。
AI 中文摘要
在有向图上为多个智能体寻找最优会合点是网络分析、运筹学和计算几何中研究的一个经典问题。在本工作中，我们利用文献中考察的最优会合点的两个目标来研究两种相遇距离的概念：$d^{\max}(u,v)$，即对所有会合点 $w$ 取 $\max(d(u,w), d(v,w))$ 的最小值；以及 $d^+(u,v)$，即对所有会合点取 $d(u,w) + d(v,w)$ 的最小值。这些值衡量了两个智能体相遇所需的最短时间和最小总距离。我们开启了在两种相遇距离概念下基本图参数（即直径、半径和离心率）的细粒度研究。对于一般有向图，我们给出了一个 $\tilde{O}(m\sqrt{n})$ 时间的算法来计算两种相遇直径的 2 近似，并证明在 SETH 假设下该结果是最优的。相反，我们表明对于相遇半径无法获得这样的结果，因为在 Hitting Set 猜想下，任何有限近似都需要二次时间。对于有向无环图，我们获得了更强的结果。我们在线性时间内精确计算 meet$^{\max}$-直径，并给出了 meet$^{+}$-直径的线性时间 2 近似。我们通过 SETH 假设下任何 $(3/2-\varepsilon)$ 近似的二次时间下界来补充后者，从而在两种相遇距离目标之间产生了分离。最后，我们研究了 $k$ 元组顶点的广义相遇距离。对于每个正整数 $\ell$，我们将 $k$ 点相遇直径的 $\ell$ 近似问题归约为在更小的元组上计算精确相遇直径，从而在时间 \\[ O\\!\left( mn+ \ell \left\lceil k^{1/\ell}\right\rceil n^{\left\lceil k^{1/\ell}\right\rceil+1} \right) \\] 内获得 $\ell$ 近似。
英文摘要
Finding an optimal meeting point for a collection of agents on a directed graph is a classical problem studied in the context of network analysis, operations research and computational geometry. In this work, we use the two objectives of optimal meeting points examined in the literature to study two notions of meet-distance: $d^{\max}(u,v)$, the minimum over all meeting points $w$ of $\max(d(u,w), d(v,w))$; and $d^+(u,v)$, the minimum over all meeting points of $d(u,w) + d(v,w)$. These values measure the minimum time and minimum total distance required for two agents to meet. We initiate the fine-grained study of fundamental graph parameters under the two notions of meet-distance, namely the diameter, radius and eccentricities. For general directed graphs, we give an $\tilde{O}(m\sqrt{n})$ time algorithm for computing a 2-approximation to both notions of meet-diameter and show that this result is optimal under SETH. In contrast, we show that such a result is unattainable for the meet-radius as any finite approximation requires quadratic time under the Hitting Set Conjecture. For directed acyclic graphs, we obtain stronger results. We compute the meet$^{\max}$-diameter exactly in linear time and give a linear-time $2$-approximation for the meet$^{+}$-diameter. We complement the latter with a quadratic-time lower bound for any $(3/2-\varepsilon)$-approximation under SETH, yielding a separation between the two meet-distance objectives. Finally, we study the generalized meet-distance of $k$-tuples of vertices. For every positive integer $\ell$, we reduce the problem of $\ell$-approximating the $k$-point meet-diameter to computing an exact meet-diameter on smaller tuples, obtaining an $\ell$-approximation in time \[ O\!\left( mn+ \ell \left\lceil k^{1/\ell}\right\rceil n^{\left\lceil k^{1/\ell}\right\rceil+1} \right). \]
Comments20 pages
