We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-R\"{a}cke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

翻译：我们首次提出了一个总时间近乎线性的算法，用于判定在一个经历递减更新（即边删除、容量减少和费用增加）的、具有边费用和容量的有向图中，是否仍存在费用不超过 $F$ 的流。这意味着我们可以在这样的递减图上，以近乎线性的时间近似计算最小费用流的值以及 $s$-$t$ 距离。我们的框架还允许我们确定性地在近乎线性时间内维护递减图的强连通分量。这些算法在随机化和确定性两种设置下，均改进了当前已知的静态计算最小费用流的最佳运行时间。我们通过采用对偶视角来获得这些算法，这产生了基于割的算法。更准确地说，我们的算法通过求解一系列 $m^{1+o(1)}$ 个动态最小比率割问题来计算流，该问题是动态最小比率圈问题的对偶模拟，而后者是近期快速最小费用流算法的基础。我们的主要技术贡献是一个新的数据结构，它通过维护一个树割稀疏化器，能在摊还 $m^{o(1)}$ 时间内返回一个近似最优的最小比率割。这是通过设计一种新算法来实现的，该算法能够维护 [Goranci-Räcke-Saranurak-Tan, SODA 2021] 提出的动态扩展图层次结构，并且同样适用于带容量的图。我们所有的算法都是确定性的，尽管它们可以通过随机化技术进一步加速，同时仍能应对自适应敌手。