We give the first almost-linear time algorithms for several problems in incremental graphs including cycle detection, strongly connected component maintenance, $s$-$t$ shortest path, maximum flow, and minimum-cost flow. To solve these problems, we give a deterministic data structure that returns a $m^{o(1)}$-approximate minimum-ratio cycle in fully dynamic graphs in amortized $m^{o(1)}$ time per update. Combining this with the interior point method framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time algorithm for deciding the first update in an incremental graph after which the cost of the minimum-cost flow attains value at most some given threshold $F$. By rather direct reductions to minimum-cost flow, we are then able to solve the problems in incremental graphs mentioned above. At a high level, our algorithm dynamizes the $\ell_1$ oblivious routing of Rozho\v{n}-Grunau-Haeupler-Zuzic-Li (STOC 2022), and develops a method to extract an approximate minimum ratio cycle from the structure of the oblivious routing. To maintain the oblivious routing, we use tools from concurrent work of Kyng-Meierhans-Probst Gutenberg which designed vertex sparsifiers for shortest paths, in order to maintain a sparse neighborhood cover in fully dynamic graphs. To find a cycle, we first show that an approximate minimum ratio cycle can be represented as a fundamental cycle on a small set of trees resulting from the oblivious routing. Then, we find a cycle whose quality is comparable to the best tree cycle. This final cycle query step involves vertex and edge sparsification procedures reminiscent of previous works, but crucially requires a more powerful dynamic spanner which can handle far more edge insertions. We build such a spanner via a construction that hearkens back to the classic greedy spanner algorithm.

翻译：我们针对增量图中的若干问题首次给出了近线性时间算法，包括环检测、强连通分量维护、$s$-$t$最短路径、最大流以及最小费用流。为解决这些问题，我们提出了一种确定性数据结构，可在全动态图中以均摊$m^{o(1)}$时间返回$m^{o(1)}$近似的最小比率环。结合Brand-Liu-Sidford（STOC 2023）的内点法框架，该算法首次实现了增量图中首个更新后最小费用流代价不超过给定阈值$F$的判定问题。通过直接归约到最小费用流问题，我们进而解决了上述增量图问题。在高层次上，我们的算法将Rozho\v{n}-Grunau-Haeupler-Zuzic-Li（STOC 2022）的$\ell_1$遗忘路由技术动态化，并发展了一种从遗忘路由结构中提取近似最小比率环的方法。为维护遗忘路由，我们采用了Kyng-Meierhans-Probst Gutenberg的并行工作工具——该工作设计了最短路径顶点稀疏化器以维护全动态图中的稀疏邻域覆盖。在环搜索方面，我们首先证明近似最小比率环可表示为遗忘路由产生的少量树上的基本环，进而寻找质量可与最优树环比拟的环。最终环查询步骤涉及类似于先前工作的顶点与边稀疏化过程，但关键需要一种能处理更多边插入的更强动态伸展子图。我们通过回溯经典贪心伸展子图算法的构造实现了这种伸展子图。