We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM '98]. Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

翻译：我们给出一个确定性 $m^{1+o(1)}$ 时间算法，用于计算具有 $m$ 条边且需求、代价和容量均为多项式有界整数有向图上的精确最大流和最小费用流。由此，自 Goldberg-Rao [J.ACM '98] 的工作以来，我们首次在计算多项式有界容量图最大流的确定性算法运行时间上取得改进。我们的算法建立在 Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] 的框架之上，该框架通过计算一系列 $m^{1+o(1)}$ 近似无向最小比环来求解最优流。我们开发了一种确定性动态图数据结构，以均摊 $m^{o(1)}$ 时间每次边更新的代价计算这样一系列最小比环。我们的关键技术贡献是 Chen 等人数据结构的顶点稀疏化和边稀疏化组件的确定性类比。对于顶点稀疏化组件，我们给出一种避免 Chen 等人中随机性的方法，该方法涉及采样随机树进行递归。对于边稀疏化组件，我们设计了一种确定性算法，该算法维持动态图向稀疏度数图的嵌入。我们还展示了如何将我们的动态度数图应用于给定一种确定性数据结构，该数据结构在边长多项式有界的图上维持一个完全动态的低拉伸生成树，具有次多项式平均拉伸和次多项式均摊时间每次边更新。