Abstract
Given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings. The standard algorithm to calculate a minimum capacity cut, due to Gomory and Hu (1961), runs in O(n 4) time and is difficult to implement. We present an alternative algorithm with the same worst-case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm. We report computational results for graphs with up to 2000 nodes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Crowder and M. Padberg, “Solving large-scale symmetric travelling salesman problems to optimality,”Management Science 26 (1980) 495–509.
P. Elias, A. Feinstein and C.E. Shannon, “Note on maximum flow through a network,”Institute of Radio Engineers, Transactions on Information Theory IT-2 (1956) 117–119.
L.R. Ford Jr. and D.R. Fulkerson, “Maximal flow through a network,”Canadian Journal of Mathematics 8 (1956) 399–404.
A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”Proceedings of the Eight Annual ACM Symposium on the Theory of Computing (Berkeley, CA, 1986) pp. 136–146.
R.E. Gomory and T.C. Hu, “Multi-terminal network flows,”SIAM Journal on Applied Mathematics 9 (1961) 551–570.
R.M. Karp, “The probabilistic analysis of combinatorial optimization algorithms,” Contributed paper,Tenth International Symposium on Mathematical Programming (Montreal, August 1979).
T.G. Lewis and W.H. Payne, “Generalized feedback shift register pseudorandom number algorithm,”Journal of the Association for Computing Machinery 20 (1973) 456–468.
M. Padberg and G. Rinaldi, “Optimization of a 532-city symmetric traveling salesman problem,”Operations Research Letters 6 (1987) 1–7.
M. Padberg and G. Rinaldi, “An efficient algorithm for the minimum capacity cut problem,” Research Report R.222, IASI-CNR (Rome, 1988a).
M. Padberg and G. Rinaldi, “A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems,” Research Report R.247, IASI-CNR (Rome, 1988b).
M. Padberg and L.A. Wolsey, “Trees and cuts,” in: C. Berge et al., eds.,Combinatorial Mathematics, Annals of Discrete Mathematics 17 (North-Holland, Amsterdam, 1983) pp. 511–517.
J.C. Picard and M. Queyranne, “Selected applications of minimum cuts in networks,”INFOR 20 (1982) 394–422.
G. Rinaldi and M. Padberg, “An efficient algorithm for the minimum capacity cut problem in large sparse graphs,” Research Report R.166, IASI-CNR (Rome, 1986).
R.E. Tarjan,Data Structure and Network Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983).
Author information
Authors and Affiliations
Additional information
Partial financial support by NSF grant DMS8508955 and ONR grant R&T4116663.
Work done while visiting New York University. Partial financial support by a New York University Research Challenge Fund grant and ONR grant R&T4116663.
Rights and permissions
About this article
Cite this article
Padberg, M., Rinaldi, G. An efficient algorithm for the minimum capacity cut problem. Mathematical Programming 47, 19–36 (1990). https://doi.org/10.1007/BF01580850
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580850