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

  • Authors:

  • Mateo Restrepo;David P. Williamson

  • Affiliations:

  • Cornell University, Ithaca, NY;Cornell University, Ithaca, NY

  • Venue:
  • SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
  • Year:
  • 2006

Quantified Score

Hi-index 0.00

Visualization

Abstract

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 [26] 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 [27]. Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne [27]) 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 (mBε)) 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(m3n log(mB)). The fastest known running time for this problem is Õ(m2n log B), and is due to Radzik [22], building on earlier work of Goldfarb, Jin, and Orlin [14].