基于自适应预处理的几乎线性时间流
Flows in Almost Linear Time via Adaptive Preconditioning
摘要：
我们提出了一个大类的流动和回归问题的解决算法，单位加权图的$(1 + 1 / poly(n))$精度在几乎线性的时间。这些问题包括$\ell_p$ -范数最小化流的$p$大（$p \in [ω(1), o(\log^{2/3} n) ]$），和他们的双，$\ell_p$ -范数半监督学习的$p$接近$1$。当$p$趋于无穷大时，$\ell_p$ -范数流及其对偶分别趋于最大流和最小割.使用这种连接和我们的算法，我们给出了一个替代的方法来近似无向最大流，和第一次几乎线性时间近似的离散化的总变差最小化目标。该算法表明，许多工具以前被认为是有限的线性系统，实际上适用于更广泛的凸目标。它基于Spielman和Teng（STOC '04，SIMAX '14）的Laplacian线性系统的基于路由的求解器，但需要几个新的工具：自适应非线性预处理，基于树路由的混合$\ell_2$和$\ell_p$范数目标的超稀疏化，以及将图分解为均匀扩展器。
英文摘要：
We present algorithms for solving a large class of flow and regression problems on unit weighted graphs to $(1 + 1 / poly(n))$ accuracy in almost-linear time. These problems include $\ell_p$-norm minimizing flow for $p$ large ($p \in [ω(1), o(\log^{2/3} n) ]$), and their duals, $\ell_p$-norm semi-supervised learning for $p$ close to $1$. As $p$ tends to infinity, $\ell_p$-norm flow and its dual tend to max-flow and min-cut respectively. Using this connection and our algorithms, we give an alternate approach for approximating undirected max-flow, and the first almost-linear time approximations of discretizations of total variation minimization objectives. This algorithm demonstrates that many tools previous viewed as limited to linear systems are in fact applicable to a much wider range of convex objectives. It is based on the the routing-based solver for Laplacian linear systems by Spielman and Teng (STOC '04, SIMAX '14), but require several new tools: adaptive non-linear preconditioning, tree-routing based ultra-sparsification for mixed $\ell_2$ and $\ell_p$ norm objectives, and decomposing graphs into uniform expanders.
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