Mathematical Programming

, Volume 118, Issue 1, pp 47–74 | Cite as

A simple GAP-canceling algorithm for the generalized maximum flow problem

  • Mateo Restrepo
  • David P. Williamson


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(m 2 log (mB)) iterations to compute an optimal flow, for an overall running time of O(m 3 nlog(mB)). The fastest known running time for this problem is \(\tilde{O}(m^2n\log B)\) , 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).

Mathematics Subject Classification (2000)

68Q25 90C35 90B10 05C85 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA
  2. 2.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

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