更快的发散最大化以获得更快的最大流
Faster Divergence Maximization for Faster Maximum Flow
摘要：
本文给出了一个算法，该算法给出了任意$m$ -边$n$ -顶点有向图，其容量至多为整数$U$，计算任意顶点$s$和$t$在$m^{4/3+o(1)}U^{1/3}$时间内的最大$s$-$t$流.当图不是太密集或具有大容量时，这改进了$m^{11/8+o(1)}U^{1/4}$（Liu Sidford 2019）、$\tilde{O}(m \sqrt{n} \log U)$（Lee Sidford 2014）和$O(mn)$（Orlin 2013）之前的最佳运行时间。为了实现本文的结果，我们建立在以前的算法方法，最大流内点法（IPM）的基础上。特别是，我们克服了先前在最大流IPM中取得的进展的关键瓶颈（Mackdry 2013，Mackdry 2016，Liu Sidford 2019），通过最大化局部$\ell_2$范数的能量最小化电流来取得进展。我们推广这种方法，而不是最大限度地减少布雷格曼发散距离加权对数障碍的流量的发散。这使得我们的算法可以避免依赖于其他IPM框架中出现的$\ell_4$规范（例如Cohen Mendry Sankowski Vladu 2017，Axiotis Mendry Vladu 2020）。此外，我们还证明了平滑的$\ell_2$ - $\ell_p$流（Kyng，Peng，Sachdeva，Wang 2019），我们以前用来有效地最大化能量（Liu Sidford 2019），也可以用来有效地最大化发散，从而产生我们想要的运行时间。我们相信这两个广义的能量最大化和广义流求解器，我们开发的可能会进一步感兴趣。
英文摘要：
In this paper we provide an algorithm which given any $m$-edge $n$-vertex directed graph with integer capacities at most $U$ computes a maximum $s$-$t$ flow for any vertices $s$ and $t$ in $m^{4/3+o(1)}U^{1/3}$ time. This improves upon the previous best running times of $m^{11/8+o(1)}U^{1/4}$ (Liu Sidford 2019), $\tilde{O}(m \sqrt{n} \log U)$ (Lee Sidford 2014), and $O(mn)$ (Orlin 2013) when the graph is not too dense or has large capacities. To achieve the results this paper we build upon previous algorithmic approaches to maximum flow based on interior point methods (IPMs). In particular, we overcome a key bottleneck of previous advances in IPMs for maxflow (Mądry 2013, Mądry 2016, Liu Sidford 2019), which make progress by maximizing the energy of local $\ell_2$ norm minimizing electric flows. We generalize this approach and instead maximize the divergence of flows which minimize the Bregman divergence distance with respect to the weighted logarithmic barrier. This allows our algorithm to avoid dependencies on the $\ell_4$ norm that appear in other IPM frameworks (e.g. Cohen Mądry Sankowski Vladu 2017, Axiotis Mądry Vladu 2020). Further, we show that smoothed $\ell_2$-$\ell_p$ flows (Kyng, Peng, Sachdeva, Wang 2019), which we previously used to efficiently maximize energy (Liu Sidford 2019), can also be used to efficiently maximize divergence, thereby yielding our desired runtimes. We believe both this generalized view of energy maximization and generalized flow solvers we develop may be of further interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
