Decremental All-Pairs Shortest Paths in Deterministic Near-Linear Time

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 $O\left (m^{{1+O(\epsilon)}\cdot} (\log m)^{{O(1/\epsilon2)}\cdot} \log L\right )$.