单位容量图中快速最小费用流的循环控制
Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs
摘要：
本文提出了一个求解单位容量图中最小费用流问题的$m^{4/3+o(1)}\log W$ -时间算法，其中$W$是任意边权的最大绝对值.对于稀疏图，这改进了该问题的最佳已知运行时间，并且通过众所周知的减少，还意味着改进了具有负权重的最短路径问题的运行时间，当$\|\boldsymbol{\mathit{b}}\|_1 = O(m)$时最小成本二分$\boldsymbol{\mathit{b}}$匹配，并恢复了当前最快算法的运行时间，用于具有单位容量的图中的最大流（Liu-Sidford，2020）。我们的算法依赖于开发一个基于内点方法的框架，该框架作用于底层图中的循环空间。从组合的角度来看，这个框架可以被看作是迭代地提高成本的次优解决方案，推动周围的流通。这些循环是通过计算标准牛顿步骤的正则化版本导出的，该版本部分受到先前关于单位容量最大流问题的工作的启发（Liu-Sidford，2019），随后根据该问题的最新进展进行了改进（Liu-Sidford，2020）。然后，可以使用最近关于$\ell_p$范数最小化流的工作来有效地计算由此产生的步骤问题（Kyng-Peng-Sachs-Wang，2019）。我们得到我们的更快的算法相结合，这个新的步骤原语与定制的预处理方法，其目的是确保这些循环计算的图形有足够大的电导。
英文摘要：
We present an $m^{4/3+o(1)}\log W$-time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where $W$ is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite $\boldsymbol{\mathit{b}}$-matching when $\|\boldsymbol{\mathit{b}}\|_1 = O(m)$, and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method-based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very recent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on $\ell_p$-norm minimizing flows (Kyng-Peng-Sachdeva-Wang, 2019). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance.
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