Generalized Preconditioning and Undirected Minimum-Cost Flow
Generalized Preconditioning and Undirected Minimum-Cost Flow
复制标题
广义预处理和无向最小成本流
DOI：
10.1137/1.9781611974782.49
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Jonah Sherman
中科院分区：
文献类型：
--
作者：
Jonah Sherman
我们提供了一个几乎线性的时间近似算法，用于在无方面的图表中无影响的最低成本流，以及一个更通用的框架，用于求解该形式的问题：找到x满足x的x满足ax = a ax = b，其中最小的norm || x ||规范通常是非欧盟人。 ax≈b明显超出了数值精度，另一方面，现有的几乎是线性的时间求解器使用双重或惩罚方法AX ||。经典条件编号遵循我们的框架，算法设计器的任务减少到设计广义的预处理。无向图带有成本标记的m边缘，并标记为需求的n个顶点，需要ϵ −2m1 + o（1） - 时间，并输出A流量以总成本（1 + ϵ）乘以总成本（1 +）大于最小的，以及提供几乎急诊的双重解决方案。通过将未对准的坐标随机舍入的晶格对相应的预处理的一般性状况分析。用于晶格拉普拉斯系统。
We present a nearly-linear time approximation algorithm for uncapacitated minimum-cost flow in undirected graphs, along with a more general framework for approximately solving problems of the form: find x satisfying Ax = b with minimal norm ||x||, where the norm is generally non-Euclidean. For most of the extensive applications of the latter problem, the exact constraints are essential, so an x satisfying Ax = b with almost-minimal norm is acceptable, while relaxing Ax = b to Ax ≈ b significantly beyond numerical precision is not. On the other hand, existing nearly-linear time solvers for non-Euclidean norms use dual or penalty methods, yielding the opposite notion where ||x|| is minimal while ||b − Ax|| ≤ t−Ω(1) after t iterations. We show that by composing solvers of the latter type, we may obtain solvers of the more-useful former type. Convergence of the composed solvers depends strongly on a generalization of the classical condition number to general norms. Following our framework, the task of the algorithm designer for such problems is reduced to that of designing a generalized preconditioner for A. Applying the framework to uncapacitated minimum-cost flow, we present an algorithm that, given an undirected graph with m edges labelled with costs, and n vertices labelled with demands, takes ϵ−2m1+o(1)-time and outputs a flow routing the demands with total cost at most (1 + ϵ) times larger than minimal, along with a dual solution proving near-optimality. The generalized preconditioner is obtained by embedding the cost metric into l1, and then considering a simple hierarchical routing scheme in l1 where demands initially supported on a dense lattice are pulled from a sparser lattice by randomly rounding unaligned coordinates to their aligned neighbors. Analysis of the generalized condition number for the corresponding preconditioner follows that of the classical multigrid algorithm for lattice Laplacian systems.