通过内点算法实现更快的近似有损广义流
Faster Approximate Lossy Generalized Flow via Interior Point Algorithms
AI总结：
提出一种基于内点算法和对称M矩阵线性系统近线性时间求解器的算法，对有损广义网络流问题实现加性ε近似，时间复杂度为Õ(m^(3/2) log(1/ε))，相比先前算法改进约m^(1/2)因子。
摘要：
我们提出了广义网络流问题的更快近似算法。广义流是指流出边的流量与流入边的流量相差一个常数因子。我们仅限于有损情况，即这些因子至多为1。我们的算法使用标准内点算法来求解网络流问题的线性规划形式。内点算法每一步产生的线性方程组形式为对称M矩阵。我们提出了一种在近线性时间内求解此类系统的算法。该算法依赖于Spielman-Teng的用于求解对角占优矩阵线性系统的近线性时间算法。对于具有m条边的图，我们的算法在时间Õ(m^(3/2) * log(1/ε))内获得最大广义流和最小费用广义流的加性ε近似。在许多参数范围内，这比先前算法改进了大约m^(1/2)因子。我们还获得了标准最小费用流问题精确求解的类似改进。
英文摘要：
We present faster approximation algorithms for generalized network flow problems. A generalized flow is one in which the flow out of an edge differs from the flow into the edge by a constant factor. We limit ourselves to the lossy case, when these factors are at most 1. Our algorithm uses a standard interior-point algorithm to solve a linear program formulation of the network flow problem. The system of linear equations that arises at each step of the interior-point algorithm takes the form of a symmetric M-matrix. We present an algorithm for solving such systems in nearly linear time. The algorithm relies on the Spielman-Teng nearly linear time algorithm for solving linear systems in diagonally-dominant matrices. For a graph with m edges, our algorithm obtains an additive epsilon approximation of the maximum generalized flow and minimum cost generalized flow in time tildeO(m^(3/2) * log(1/epsilon)). In many parameter ranges, this improves over previous algorithms by a factor of approximately m^(1/2). We also obtain a similar improvement for exactly solving the standard min-cost flow problem.
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