We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair $x,y$ of vertices of $G$, return a path $P$ connecting $x$ to $y$, whose length is within factor $\alpha$ of the length of the shortest $x$-$y$ path, in time $\tilde O(|E(P)|)$, where $\alpha$ is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees for the oblivious-adversary setting. Unfortunately, adaptive-adversary setting is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS '13, SICOMP '16] achieves a $(1+\epsilon)$-approximation with total update time $\tilde O(mn/\epsilon)$; the best current total update time of $n^{2.5+O(\epsilon)}$ is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA'21], with $2^{O(1/\epsilon)}$-multiplicative and $2^{O(\log^{3/4}n/\epsilon)}$-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time $O(n^{3}\log L/\epsilon)$, achieving a $(1+\epsilon)$-approximation. Here, L is the ratio of longest to shortest edge lengths. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any $\Omega(1/\log\log m)\leq \epsilon< 1$, achieves approximation factor $(\log m)^{2^{O(1/\epsilon)}}$, with total update time $O\left (m^{1+O(\epsilon)}\cdot (\log m)^{O(1/\epsilon^2)}\cdot \log L\right )$.

中文翻译：

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确定性近线性时间中的递减所有对最短路径

我们研究了无向边加权图中的递减 All-Pairs Shortest Paths (APSP) 问题。问题的输入是一个 $n$-顶点 $m$-边图 $G$，边长为非负，经历了一系列边删除。目标是支持近似最短路径查询：给定一对 $x,y$ 的 $G$ 顶点，返回连接 $x$ 到 $y$ 的路径 $P$，其长度在因子 $\alpha$ 内最短$x$-$y$路径的长度，时间$\tilde O(|E(P)|)$，其中$\alpha$是算法的近似因子。APSP 是最基本和最广泛研究的动态图问题之一。在 [Chechik, FOCS 2018] 的算法中，一长串的工作达到了高潮，为不经意的对手设置提供了近乎最佳的保证。不幸的是，适应性对手设置仍然知之甚少。对于未加权的图，[Henzinger, Krinninger 和 Nanongkai, FOCS '13, SICOMP '16] 的算法实现了 $(1+\epsilon)$-近似，总更新时间为 $\tilde O(mn/\epsilon)$；当前最佳总更新时间 $n^{2.5+O(\epsilon)}$ 是通过 [Chuzhoy, Saranurak, SODA'21] 的确定性算法实现的，$2^{O(1/\epsilon)}$ - 乘法和 $2^{O(\log^{3/4}n/\epsilon)}$-加法近似。据我们所知，对于任意非负边权重，当前最快的自适应更新算法的总更新时间为 $O(n^{3}\log L/\epsilon)$，实现了 $(1+\ epsilon)$-近似值。其中，L 是最长边与最短边长度的比值。我们的主要结果是无向边加权图中递减 APSP 的确定性算法，即，对于任何 $\Omega(1/\log\log m)\leq \epsilon< 1$，