### G-2013-52

# About the Minimum Mean Cycle-Canceling Algorithm

## Jean-Bertrand Gauthier, Jacques Desrosiers, and Marco E. Lübbecke

This paper focuses on the resolution of the capacitated minimum cost flow problem on a network comprising *n* nodes and *m* arcs. We present a method that counts imperviousness to degeneracy among its strengths, namely the *minimum mean cycle-canceling* algorithm (MMCC). At each iteration, primal feasibility is maintained and the objective function strictly improves. The goal is to write a uniform and hopefully more accessible paper which centralizes the ideas presented in the seminal work of Goldberg and Tarjan (1989) as well as the additional paper of Radzik and Goldberg (1994) where the complexity analysis is refined.
Important properties are proven using linear programming rather than constructive arguments.

We also retrieve Cancel-and-Tighten from the former paper, where each so-called phase which can be seen as a group of iterations requires *O*(*m* log *n*) time. MMCC turns out to be a strongly polynomial algorithm which runs in *O*(*m**n*) phases, hence in *O*(*m*^{2n} log *n*) time. This new complexity result is obtained with a combined analysis of the results in both papers along with original contributions which allows us to enlist Cancel-and-Tighten as an acceleration strategy.

Published **September 2013**
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28 pages

This cahier was revised in **July 2014**