最大流算法统计

对最大流算法历史文献的一个调研

Table: Polynomial algorithms for the max flow problem

No	Duo to	Year	Running Time;
1	Ford & Fulkerson [1]	1956	\(O(nmU)\)
2	Edmonds and Karp [2]	1972	\(O(nm^2)\)
3	Dinic [3]	1970	\(O(n^2m)\)
4	Karzanov [4]	1974	\(O(n^3)\)
5	Cherkasky [5]	1977	\(O(n^2\sqrt{m})\)
6	Malhotra, Kumar & Maheshwari [6]	1977	\(O(n^3)\)
7	Galil [7]	1980	\(O(n^(5/3)m^(2/3))\)
8	Galil & Naaman [8]	1980	\(O(nmlog^2n)\)
9	Sleator & Tarjan [9]	1983	\(O(nmlogn)\)
10	Gabow [10]	1985	\(O(nmlogU)\)
11	Goldberg & Tarjan [11]	1988	\(O(nmlog(n^2/m))\)
12	Ahuja & Orlin [12]	1989	\(O(nm + n^2logU)\)
13	Ahuja, Orlin & Tarjan [13]	1989	\(O(nmlog(n\sqrt{U}/(m + 2))\)
14	King, Rao & Tarjan [14]	1992	\(O(nm+n^{2+e})\)
15	King, Rao & Tarjan [15]	1994	\(O(nmlog_{m/nlogn}n)\)
16	Cheriyan, Hagerup & Mehlhorn [16]	1996	\(O(n^3/logn)\)
17	Goldberg & Rao [17]	1998	\(O(min\{n^(2/3),m^{1/2}\}mlog{n2/m}logU)\)
18	Orlin [18]	2012	\(O(nm)\)
19	Orlin [18]	2012	\(O(n^2/logn) if m = O(n)\)

• [1] L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956.
• [2] J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic eciency for network flow problems. Journal of the ACM, 19:248-264, 1972.
• [3] E. A. Dinic. Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Mathematics Doklady, 11:1277{1280, 1970
• [4] A. V. Karzanov. Determining the maximal flow in a network by the method of preflows. Soviet Mathematics Doklady, 15:434-437, 1974.
• [5] B. V. Cherkasky. Algorithm for construction of maximal flow in networks with complexity of \(O(V^2\sqrt{E})\) operations. Mathematical Methods of Solution of Economical Problems, 17:112-125, 1977. (In Russian).
• [6] V. M. Malhotra, P. Kumar, and S. N. Maheshwari. An \(O(V^3)\) algorithm for finding the maximum flows in networks. Information Processing Letters, 7:277-278, 1978.
• [7] Z. Galil. An \(O(V^{5/3}E^{2/3})\) algorithm for the maximal flow problem. Acta Informatica, 14(3):221-242, 1980.
• [8] Z. Galil and A. Naaman. An \(O(VElog^2E)\) algorithm for the maximal flow problem. J.Computer and System Sciences, 21:203-217., 1980.
• [9] D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees. J. Computer and System Sciences, 24:362-391, 1983.
• [10] H. N. Gabow. A data structure for dynamic trees. J. Computer and System Sciences, 31:148-168, 1985.
• [11] A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem. Journal of the ACM, 35:921-940, 1988.
• [12] R. K. Ahuja and J. B. Orlin. A fast and simple algorithm for the maximum flow problem. Operations Research, 37:748-759, 1989.
• [13] R. K. Ahuja, J. B. Orlin, and R. E. Tarjan. Improved time bounds for the maximum flow problem. SIAM Journal on Computing, 18:939-954, 1989.
• [14] V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 157{164, 1992.
• [15] V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. J. Algorithms, 23:447-474, 1994.
• [16] J. Cheriyan, T. Hagerup, and K. Mehlhorn. An \(O(n^3)\) time maximum-flow algorithm. SIAM Journal on Computing, 45:1144-1170, 1996.
• [17] A. V. Goldberg and S. Rao. Beyond the flow decomposition barrier. Journal of the ACM, 45:783-797, 1998.
• [18] J. B. Orlin, “Max flows in \(o(nm)\) time, or better,” in Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, ser. STOC ’13. New York, NY,USA: ACM, 2013, pp. 765–774. [Online]. Available: http://doi.acm.org/10.1145/2488608.2488705