一个通过距离预言机的最小比率环实现的更简单、更快的最小费用流求解器
A Simpler and Faster Min-Cost Flow Solver via Min-Ratio Cycles from Distance Oracles
AI总结：
本文通过直接从动态距离预言机提取最小比率环，简化了最小费用流算法，得到更快的原始最大流和最小费用流求解器，并支持增量图。
AI中文摘要：
Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva（FOCS 2022）提出的首个近线性时间最大流和最小费用流算法，将这些流目标归约为一系列最小比率环问题。解决这一核心原语需要近似最小化线性梯度项与无向长度项之比。在Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva（FOCS 2022）及其后续工作Chen-Kyng-Liu-Meierhans-Probst Gutenberg（STOC 2024）中，给出了复杂的数据结构来解决最小比率问题。我们证明，利用线性性，可以直接从Kyng-Meierhans-Probst Gutenberg（STOC 2024）的动态距离预言机中提取这样的环。这简化了先前依赖多个额外步骤来提取环的算法，并可视为证据表明解决最小比率环问题确实完全关乎距离。因此，我们获得了一个更快的原始最大流和最小费用流求解器，并且该求解器还扩展到增量图。
英文摘要：
The first almost-linear time maximum and minimum cost flow algorithm of Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022), reduced these flow objectives to a sequence of min-ratio cycle problems. Solving this core primitive requires approximately minimizing the ratio of a linear gradient term and an undirected length term. In Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022) and the subsequent work of Chen-Kyng-Liu-Meierhans-Probst Gutenberg (STOC 2024), intricate data structures were given to solve the min-ratio problem. We show that such a cycle can be extracted directly from the dynamic distance oracle of Kyng-Meierhans-Probst Gutenberg (STOC 2024) using linearity. This simplifies previous algorithms that relied on multiple additional steps to extract the cycle, and can be seen as evidence that solving the min-ratio cycle problem really is all about distances. As a result, we obtain a faster primal maxflow and min-cost flow solver that also extends to incremental graphs.
