A simple GAP-canceling algorithm for the generalized maximum flow problem
A simple GAP-canceling algorithm for the generalized maximum flow problem
复制标题
广义最大流问题的简单 GAP 消除算法
DOI：
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发表时间：
2006
期刊：
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通讯作者：
David P. Williamson
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文献类型：
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作者：
Mateo Restrepo;David P. Williamson
我们为一般最大流量问题提供了一种简单的主要算法，该算法反复地发现并取消了广义增强路径（GAPS）。在差距的增加与其弧的剩余能力之间取决于良好的权衡；我们的算法可能被视为韦恩的算法的特殊情况–459，2002）。每个不等式的变量（TVPIS）。 O（Mn）计算ε-最佳流量的时间，或O（M2 log（MB））迭代以计算O最佳流量，以o（M3NLOG（MB））的整体运行时间。这个问题是$$ ILDE {o}（M^2nlog b）$$，并且是由于Radzik（Theor Comput Sci 312：75–97，2004），基于Goldfarb等人的早期工作。 ：793–802，1997）。
We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O(mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O(m log(mB/ε)) iterations of O(mn) time to compute an ε-optimal flow, or O(m2 log (mB)) iterations to compute an optimal flow, for an overall running time of O(m3nlog(mB)). The fastest known running time for this problem is $$ ilde{O}(m^2nlog B)$$ , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997).