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DOI: 10.7155/jgaa.00539
Smoothed Analysis of the MinimumMean Cycle Canceling Algorithm and the Network Simplex Algorithm
Kamiel Cornelissen and
Bodo Manthey
Vol. 24, no. 3, pp. 397421, 2020. Regular paper.
Abstract The minimumcost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood.
Over the last half century many algorithms have been developed to solve the MCF problem, and these algorithms have varying worstcase bounds on their running time. However, these worstcase bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is among the best algorithms in practice and performs very well in experimental studies. On the other hand, the MinimumMean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies.
To explain these differences in performance in practice we apply the framework of smoothed analysis to NS and MMCC. We show an upper bound of $O(mn^2\log(n)\log(\phi))$ for the number of iterations of the MMCC algorithm. Here, $n$ is the number of nodes, $m$ is the number of edges, and $\phi$ is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of $\Omega(m\log(\phi))$ for the number of iterations of the MMCC algorithm, which can be strengthened to $\Omega(mn)$ when $\phi=\Theta(n^2)$. For the number of iterations of the NS algorithm we show a smoothed lower bound of $\Omega(m \cdot \min \{ n, \phi \} \cdot \phi)$.

Submitted: March 2019.
Reviewed: June 2020.
Revised: July 2020.
Accepted: July 2020.
Final: July 2020.
Published: July 2020.
Communicated by
Michael Kaufmann

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