稀疏网络和稠密网络上的最大流
Max-Flows on Sparse and Dense Networks
摘要：
本文对具有$n$顶点和$m$弧的一般网络的最大流问题提出了一种改进算法。我们展示了如何在$O(mn)$时间（当$m = O(n^{2-ε})$时）解决某些$0 <ε\leq 1$的问题。这改善了结果的Orlin和国王，等。例如，分别在$O(mn + m^{31/16} \log^2 n)$和$O(mn\log_{m/n\log n}n)$时间内解决了问题。我们的主要结果是在所有缩放阶段减少非饱和推送到$O(mn)$的数量。我们的算法可以被看作是补充国王，等。例如，在这个意义上，我们解决了$O(mn)$时间的最大流问题时，$m = O(n^{2-ε})$（所有稀疏和非密集网络），而国王，et.当$m = Ω(n^{1+ε})$（所有密集和非稀疏网络）时，在$O(mn)$时间内解决它。我们的改进是通过Ahuja和Orlin的超额标度方法和Orlin的紧致流网络的新组合来实现的。据我们所知，这是第一个运行在这一系列网络上的$O(mn)$时间最大流算法。进一步，我们将Orlin的$O(mn)$时间算法的范围从$O(n^{16/15-ε})$扩展到$O(n^{2-ε})$，这是对近似$O(n^{0.94})$的改进.我们的研究结果还表明，该问题可以解决所有的$n$和$m$专门使用推重新标记的方法。我们还给出了参数流的改进算法和有效地构造Gomory-Hu树，并提出了一种新的方法来解决最小费用流问题。
英文摘要：
In this paper, we present an improved algorithm for the maximum flow problem on general networks with $n$ vertices and $m$ arcs. We show how to solve the problem in $O(mn)$ time, when $m = O(n^{2-ε})$, for some $0 <ε\leq 1$. This improves upon the results of both Orlin and King, et. al., who solved the problem in $O(mn + m^{31/16} \log^2 n)$ and $O(mn\log_{m/n\log n}n)$ time, respectively. Our main result is reducing the number of nonsaturating pushes to $O(mn)$ across all scaling phases. Our algorithm can be seen as complementary to King, et. al., in the sense that we solve the max-flow problem in $O(mn)$ time when $m = O(n^{2-ε})$ (all sparse and non-dense networks), whereas King, et. al. solve it in $O(mn)$ time when $m = Ω(n^{1+ε})$ (all dense and non-sparse networks). Our improvement is reached by a novel combination of Ahuja and Orlin's excess scaling method and Orlin's compact flow networks. To our knowledge, this is the first $O(mn)$ time max-flow algorithm that runs on this range of networks. Further, we extend the range of Orlin's $O(mn)$ time algorithm from $O(n^{16/15-ε})$ to $O(n^{2-ε})$, which is an improvement of approximately $O(n^{0.94})$. Our result also establishes that the problem can be solved for all $n$ and $m$ using exclusively the push-relabel method. We also give improved algorithms for parametric flows and efficiently constructing Gomory-Hu trees, and suggest a new approach to the minimum-cost flow problem.
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