AI 中文总结
针对加权无向图中的n对最短路径问题，提出首个(2-α)k-近似算法（常数α>0），运行时间为Õ(mn^{1/k}+n^{1+2/k})，并引入重边技术将依赖最重边权重的近似转化为纯乘法近似。
AI 中文摘要
设 $G = (V, E)$ 是一个图，其中 $n = |V|$ 个节点，$m = |E|$ 条边。由 Cohen [FOCS'93; SICOMP'99] 引入的 $t$ 对最短路径问题要求近似计算 $t$ 个指定顶点对之间的距离。最近，该问题重新受到关注，特别是在 $t = \Theta(n)$ 的情况下：即 $n$ 对最短路径问题。在此设定下，Dalirrooyfard、Jin、Vassilevska Williams 和 Wein [FOCS'22] 以及 Chechik、Hoch 和 Lifshitz [SODA'25] 开发了新的算法和条件下界。在本文中，我们提出了在\textit{加权}无向图中 $n$ 对最短路径问题的首个算法，该算法实现了 $(2 - \alpha)k$-近似（常数 $\alpha > 0$），运行时间为 $\tilde{O}(mn^{1/k} + n^{1 + 2/k})$。具体来说，我们给出了 $1.622k$-近似，改进了 Chechik、Hoch 和 Lifshitz [SODA'25] 对于非超稀疏图的 $(2k - 3)$-近似，从而肯定地回答了他们提出的开放问题。我们还针对非加权图和稠密加权图开发了具有更好折衷的改进近似算法，改进了 Dalirrooyfard 等人以及 Chechik、Hoch 和 Lifshitz 的结果。我们的主要技术贡献是新的\textit{重边}技术。利用该技术，我们将近似保证依赖于 $W_{uv}$（$u$ 和 $v$ 之间最短路径上最重边的权重）的算法转化为不依赖于 $W_{uv}$ 的纯乘法近似算法。
英文摘要
Let $G = (V, E)$ be a graph with $n = |V|$ nodes and $m = |E|$ edges. The $t$-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between $t$ prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where $t = Θ(n)$: the $n$-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25]. In this paper, we present the first algorithm for the $n$-Pairs Shortest Paths problem in \textit{weighted} undirected graphs that achieves a $(2 - α)k$-approximation, for constant $α> 0$, that runs in $\tilde{O}(mn^{1/k} + n^{1 + 2/k})$ time. Specifically, we present a $1.622k$-approximation, improving upon the $(2k - 3)$-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard \etal~and Chechik, Hoch, and Lifshitz. Our main technical contribution is the new \textit{heavy-edge} technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on $W_{uv}$, the weight of the heaviest edge on the shortest path between $u$ and $v$, into an algorithm with purely multiplicative approximation that does not depend on $W_{uv}$.
