We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate decremental APSP algorithms, two for weighted and two for unweighted graphs. Our first result is $(2+ \epsilon)$-APSP with total update time $\tilde{O}(m^{1/2}n^{3/2})$ (when $m= n^{1+c}$ for any constant $0<c<1$). Prior to our work the fastest algorithm for weighted graphs with approximation at most $3$ had total $\tilde O(mn)$ update time for $(1+\epsilon)$-APSP [Bernstein, SICOMP 2016]. Our second result is $(2+\epsilon, W_{u,v})$-APSP with total update time $\tilde{O}(nm^{3/4})$, where the second term is an additive stretch with respect to $W_{u,v}$, the maximum weight on the shortest path from $u$ to $v$. Our third result is $(2+ \epsilon)$-APSP for unweighted graphs in $\tilde O(m^{7/4})$ update time, which for sparse graphs ($m=o(n^{8/7})$) is the first subquadratic $(2+\epsilon)$-approximation. Our last result for unweighted graphs is $(1+\epsilon, 2(k-1))$-APSP, for $k \geq 2 $, with $\tilde{O}(n^{2-1/k}m^{1/k})$ total update time (when $m=n^{1+c}$ for any constant $c >0$). For comparison, in the special case of $(1+\epsilon, 2)$-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized, work against an oblivious adversary, and have constant query time.

翻译：本文针对递减全源最短路径（APSP）问题，提供了近似度与运行时间之间的新权衡。针对经历边删除操作的包含m条边和n个节点的无向图，我们提出了四种新的近似递减APSP算法，其中两种适用于加权图，两种适用于无加权图。我们的首个结果是总更新时间为$\tilde{O}(m^{1/2}n^{3/2})$的$(2+ \epsilon)$-APSP（当$m= n^{1+c}$且常数$0<c<1$时）。在此之前，加权图中近似度不超过3的最快算法是[Bernstein, SICOMP 2016]中总更新时间为$\tilde O(mn)$的$(1+\epsilon)$-APSP。第二个结果是总更新时间为$\tilde{O}(nm^{3/4})$的$(2+\epsilon, W_{u,v})$-APSP，其中第二项是基于$W_{u,v}$（即从u到v的最短路径上的最大权重）的加性偏移。第三个结果是在$\tilde O(m^{7/4})$更新时间内实现无加权图的$(2+ \epsilon)$-APSP，这对于稀疏图（$m=o(n^{8/7})$）是首个亚二次$(2+\epsilon)$近似算法。针对无加权图的最后一个结果是总更新时间为$\tilde{O}(n^{2-1/k}m^{1/k})$的$(1+\epsilon, 2(k-1))$-APSP（当$m=n^{1+c}$且常数$c >0$时，$k \geq 2$）。作为对比，在$(1+\epsilon, 2)$近似的特例中，该结果改进了[Henzinger, Krinninger, Nanongkai, SICOMP 2016]中总更新时间为$\tilde{O}(n^{2.5})$的最新算法。所有结果均为随机算法，能抵御不知情敌手攻击，并具有常数时间查询复杂度。