# Tight bounds on the number of minimum-mean cycle cancellations and related results

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## Abstract

We prove a tight Θ(min(*nm* log(nC), nm^{2})) bound on the number of iterations of the minimum-mean cycle-canceling algorithm of Goldberg and Tarjan [13]. We do this by giving the lower bound and by improving the strongly polynomial upper bound on the number of iterations to*O(nm*^{2}). We also give an improved version of the maximum-mean cut canceling algorithm of [7], which is a dual of the minimum-mean cycle-canceling algorithm. Our version of the dual algorithm runs in O(nm^{2}) iterations.

### Key words

Network flow problems Minimum cost flow Minimum cost circulation Combinatorial optimization Cycle canceling algorithms Strongly polynomial algorithms## Preview

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### References

- [1]R. K. Ahuja, A. V. Goldberg, J. B. Orlin, and R. E. Tarjan. Finding Minimum-Cost Flows by Double Scaling. Technical Report STAN-CS-88-1227, Department of Computer Science, Stanford University, 1988.Google Scholar
- [2]R. G. Busacker and T. L. Saaty.
*Finite Graphs and Networks: An Introduction with Applications*. McGraw-Hill, New York, 1965.Google Scholar - [3]W. Chi and S. Fujishige. A Primal Algorithm for the Submodular Flow Problem with Minimum-Mean Cycle Selection. Technical Report 350, University of Tsukuba, 1987.Google Scholar
- [4]J. Edmonds and R. M. Karp. Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems.
*J. Assoc. Comput. Mach.*, 19:248–264, 1972.Google Scholar - [5]G. M. Engel and H. Schneider. Diagonal Similarity and Equivalence for Matrices Over Groups with O.
*Czech. Math. J.*, 25:389–403, 1975.Google Scholar - [6]T. R. Ervolina and S. T. McCormick. A Strongly Polynomial Dual Cancel and Tighten Algorithm for Minimum Cost Network Flow. Working Paper 90-MSC-010, Faculty of Commerce, University of British Columbia, 1990.Google Scholar
- [7]T. R. Ervolina and S. T. McCormick. A Strongly Polynomial Maximum Mean Cut Cancelling Algorithm for Minimum Cost Network Flow. Working Paper 90-MSC-009, Faculty of Commerce, University of British Columbia, 1990.Google Scholar
- [8]L. R. Ford, Jr., and D. R. Fulkerson.
*Flows in Networks*. Princeton University Press, Princeton, NJ, 1962.Google Scholar - [9]D. R. Fulkerson. An Out-of-Kilter Method for Minimal Cost Flow Problems.
*SIAM J. Appl. Math*, 9:18–27, 1961.Google Scholar - [10]A. V. Goldberg. Efficient Graph Algorithms for Sequential and Parallel Computers. Ph.D. thesis, M.I.T., January 1987. (Also available as Technical Report TR-374, Laboratory for Computer Science, M.I.T., 1987).Google Scholar
- [11]A. V. Goldberg and D. Gusfield. Book Review: by G. M. Adel'son-Vel'ski, E. A. Dinic, and A. V. Karzanov. Technical Report STAN-CS-89-1313, Department of Computer Science, Stanford University, 1990.Google Scholar
- [12]A. V. Goldberg, É. Tardos, and R. E. Tarjan. Network Flow Algorithms. Technical Report STAN-CS-89-1252, Department of Computer Science, Stanford University, 1989.Google Scholar
- [13]A. V. Goldberg and R. E. Tarjan. Finding Minimum-Cost Circulations by Canceling Negative Cycles.
*J. Assoc. Comput. Mach.*, 36:873–886, 1989.Google Scholar - [14]A. V. Goldberg and R. E. Tarjan. Finding Minimum-Cost Circulations by Successive Approximation.
*Math. Oper. Res.*, 15:430–466, 1990.Google Scholar - [15]R. Hasin. Algorithms for the Minimum Cost Circulation Problem on Maximizing the Mean Improvement. Unpublished manuscript, School of Mathematical Sciences, Department of Statistics, Tel Aviv University, 1990.Google Scholar
- [16]R. Hassin. The Minimum-Cost Flow Problem: A Unifying Approach to Dual Algorithms and a New Tree-Search Algorithm.
*Math. Programming*, 25:228–239, 1983.Google Scholar - [17]K. Iwano, S. Misono, S. Tezuka, and S. Fujishige. A New Scaling Algorithm for the Maximum Mean Cut Problem.
*Algorithmica*, this issue, pp. 243–255.Google Scholar - [18]R. M. Karp. A Characterization of the Minimum Cycle Mean in a Digraph.
*Discrete Math.*, 23:309–311, 1978.Google Scholar - [19]M. Klein. A Primal Method for Minimal Cost Flows with Applications to the Assignment and Transportation Problems.
*Management Sci.*, 14:205–220, 1967.Google Scholar - [20]E. L. Lawler.
*Combinatorial Optimization: Networks and Matroids*. Holt, Reinhart, and Winston, New York, 1976.Google Scholar - [21]S. T. McCormick. A Strongly Polynomial Minimum Mean Cut Algorithm for Submodular Flow. Working Paper 90-MSC-017, Faculty of Commerce, University of British Columbia, 1990.Google Scholar
- [22]S. T. McCormick and T. R. Ervolina. Computing Maximum Mean Cuts. Working Paper 90-MSC-011, Faculty of Commerce, 1990. University of British Columbia, 1990.Google Scholar
- [23]G. J. Minty. Monotone Networks.
*Proc. Roy. Soc. London Ser. A*, 257:194–212, 1960.Google Scholar - [24]J. B. Orlin. A Faster Strongly Polynomial Minimum Cost Flow Algorithm.
*Proc. 20th Annual ACM Symposium on Theory of Computing*, pp. 377–387, 1988.Google Scholar - [25]J. B. Orlin and R. K. Ahuja. New Scaling Algorithms for Assignment and Minimum Cycle Mean Problems. Sloan Working Paper 2019-88, Sloan School of Management, M.I.T., 1988.Google Scholar
- [26]T. Radzik. Minimizing Capacity Violations in a Transshipment Network.
*Proc. 3rd ACM-SIAM Symposium on Discrete Algorithms*, pp. 185–194, 1992.Google Scholar - [27]É. Tardos. A Strongly Polynomial Minimum Cost Circulation Algorithm.
*Combinatorica*, 5(3):247–255, 1985.Google Scholar - [28]N. Zadeh. A Bad Network Problem for the Simplex Method and Other Minimum Cost Flow Algorithms.
*Math. Programming*, 5:255–266, 1973.Google Scholar - [29]N. Zadeh. More Pathological Examples for Network Flow Problems.
*Math. Programming*, 5:217–224, 1973.Google Scholar - [30]U. Zimmermann. Negative Circuits for Flows and Submodular Flows. Preprints in Optimization Series, Institute Für Angewandte Mathematik, Techniche Universität Caroto-Wilhemina zu Brauschweig, Brauschweig, 1989.Google Scholar
- [31]English transcription: G. M. Adel'son-Vel'ski, E. A. Dinic, and A. V. Karzanov,
*Potokovye Algoritmy*, Science, Moscow. Title translation: Flow Algorithms.Google Scholar - [32]English transcription: E. A. Dinic, Metod Porazryadnogo Sokrashcheniya Nevyazok i Transportnye Zadachi,
*Issledo-vaniya po Diskretnoi Matematike*, Science, Moscow. Title translation: The Method of Scaling and Transportation Problems.Google Scholar

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© Springer-Verlag New York Inc. 1994