The minimum-cost flow problem is a classic problem in combinatorial
optimization with various applications. Several pseudo-polynomial, polynomial,
and strongly polynomial algorithms have been developed in the past decades, and
it seems that both the problem and the algorithms are well understood. However,
some of the algorithms' running times observed in empirical studies contrast
the running times obtained by worst-case analysis not only in the order of
magnitude but also in the ranking when compared to each other. For example, the
Successive Shortest Path (SSP) algorithm, which has an exponential worst-case
running time, seems to outperform the strongly polynomial Minimum-Mean Cycle
Canceling algorithm.
To explain this discrepancy, we study the SSP algorithm in the framework of
smoothed analysis and establish a bound of O(mnϕ) for the number of
iterations, which implies a smoothed running time of O(mnϕ(m+nlogn)),
where n and m denote the number of nodes and edges, respectively, and
ϕ is a measure for the amount of random noise. This shows that worst-case
instances for the SSP algorithm are not robust and unlikely to be encountered
in practice. Furthermore, we prove a smoothed lower bound of Ω(mϕmin{n,ϕ}) for the number of iterations of the SSP algorithm, showing
that the upper bound cannot be improved for ϕ=Ω(n).Comment: A preliminary version has been presented at SODA 201