用增大的电流计算最大流
Computing Maximum Flow with Augmenting Electrical Flows
摘要：
本文对具有$m$弧和最大整数容量$U$的有向图的最大$s$-$t$流问题和最小$s$ - $t$割问题提出了一个$\tilde{O}\left(m^{\frac{10}{7}}U^{\frac{1}{7}}\right)$ -时间算法.在单位容量情况下，这与Mackdry（FOCS 2013）的$\tilde{O}\left((mU)^{\frac{10}{7}}\right)$时间算法的运行时间相匹配，并且在$U$适度大并且图足够稀疏时，其性能优于Mackdry（FOCS 2013）以及Lee和Sidford（FOCS 2014）的$\tilde{O}\left(m \sqrt{n} \log U\right)$时间算法。通过众所周知的减少，这也给出了类似的运行时间的最大基数二分$b$匹配问题的改善。我们的算法的优点之一是它比Madry（FOCS 2013）和Lee和Sidford（FOCS 2014）中提出的算法简单得多。特别是，这些算法采用了一个复杂的边界点方法框架，而我们的算法是直接投在经典的增广路径设置，几乎所有的组合最大流算法使用。在高层次上，所提出的算法采用原始对偶方法，其中每次迭代使用电流计算来找到当前残差图中的增强$s$ - $t$流并更新对偶解。我们表明，通过保持一定的谨慎耦合这些原始和对偶的解决方案，我们总是保证取得重大进展。
英文摘要：
We present an $\tilde{O}\left(m^{\frac{10}{7}}U^{\frac{1}{7}}\right)$-time algorithm for the maximum $s$-$t$ flow problem and the minimum $s$-$t$ cut problem in directed graphs with $m$ arcs and largest integer capacity $U$. This matches the running time of the $\tilde{O}\left((mU)^{\frac{10}{7}}\right)$-time algorithm of Mądry (FOCS 2013) in the unit-capacity case, and improves over it, as well as over the $\tilde{O}\left(m \sqrt{n} \log U\right)$-time algorithm of Lee and Sidford (FOCS 2014), whenever $U$ is moderately large and the graph is sufficiently sparse. By well-known reductions, this also gives similar running time improvements for the maximum-cardinality bipartite $b$-matching problem. One of the advantages of our algorithm is that it is significantly simpler than the ones presented in Madry (FOCS 2013) and Lee and Sidford (FOCS 2014). In particular, these algorithms employ a sophisticated interior-point method framework, while our algorithm is cast directly in the classic augmenting path setting that almost all the combinatorial maximum flow algorithms use. At a high level, the presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting $s$-$t$ flow in the current residual graph and to update the dual solution. We show that by maintain certain careful coupling of these primal and dual solutions we are always guaranteed to make significant progress.
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