Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9198)


The minimum-cost 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 worst-case bounds on their running time. However, these worst-case 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 considered the best algorithm in practice and performs best in experimental studies. On the other hand, the Minimum-Mean 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. For the number of iterations of the MMCC algorithm we show an upper bound of \(O(mn^2\log (n)\log (\phi ))\). 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 \(\Omega (m \cdot \min \{ n, \phi \} \cdot \phi )\).




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  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin. J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall (1993)Google Scholar
  2. 2.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. Journal of Computer and System Sciences 69(3), 306–329 (2004)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Brunsch, T., Cornelissen, K., Manthey, B., Röglin, H., Rösner, C.: Smoothed analysis of the successive shortest path algorithm. Computing Research Repository 1501.05493 [cs.DS], arXiv 2015. Preliminary version at SODA (2013)Google Scholar
  4. 4.
    Busacker, R.G., Gowen, P.J.: A procedure for determining a family of miminum-cost network flow patterns. Technical Report Technical Paper 15, Operations Research Office (1960)Google Scholar
  5. 5.
    Dantzig, G.B.: Linear programming and extensions. Rand Corporation Research Study. Princeton Univ. Press, Princeton (1963)Google Scholar
  6. 6.
    Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM 19(2), 248–264 (1972)MATHCrossRefGoogle Scholar
  7. 7.
    Ford, Jr. L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press (1962)Google Scholar
  8. 8.
    Fulkerson, D.R.: An out-of-kilter algorithm for minimal cost flow problems. Journal of the SIAM 9(1), 18–27 (1961)MATHGoogle Scholar
  9. 9.
    Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by canceling negative cycles. J. ACM 36(4), 873–886 (1989)Google Scholar
  10. 10.
    Iri, M.: A new method for solving transportation-network problems. Journal of the Operations Research Society of Japan 3(1,2), 27–87 (1960)Google Scholar
  11. 11.
    Jewell, W.S.: Optimal flow through networks. Operations Research 10(4), 476–499 (1962)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Karp, R.M.: A characterization of the minimum cycle mean in a digraph. Discrete Mathematics 23(3), 309–311 (1978)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Klein, M.: A primal method for minimal cost flows with applications to the assignment and transportation problems. Management Science 14(3), 205–220 (1967)MATHCrossRefGoogle Scholar
  14. 14.
    Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 1st edn. Springer Publishing Company, Incorporated (2007)Google Scholar
  15. 15.
    Kovács, P.: Minimum-cost flow algorithms: An experimental evaluation. Optimization Methods and Software 30(1), 94–127 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Manthey, B., Röglin, H.: Smoothed analysis: Analysis of algorithms beyond worst case. it - Information Technology 53(6), 280–286 (2011)Google Scholar
  17. 17.
    Minty, G.J.: Monotone networks. In Proceedings of the Royal Society of London A, pp. 194–212 (1960)Google Scholar
  18. 18.
    Orlin, J.B.: Genuinely polynomial simplex and non-simplex algorithms for the minimum cost flow problem. Technical report, Sloan School of Management. MIT, Cambridge, Technical Report No. 1615–84 (1984)Google Scholar
  19. 19.
    Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41(2), 338–350 (1993)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Orlin, J.B.: A polynomial time primal network simplex algorithm for minimum cost flows. Math. Program. 77, 109–129 (1997)MathSciNetGoogle Scholar
  21. 21.
    Radzik, T., Goldberg, A.V.: Tight bounds on the number of minimum-mean cycle cancellations and related results. Algorithmica 11(3), 226–242 (1994)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Communications of the ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  24. 24.
    Tardos, É.: A strongly polynomial minimum cost circulation algorithm. Combinatorica 5(3), 247–256 (1985)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Zadeh, N.: A bad network problem for the simplex method and other minimum cost flow algorithms. Mathematical Programming 5(1), 255–266 (1973)MATHMathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of TwenteEnschedeThe Netherlands

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