Random sampling in cut, flow, and network design problems

Mathematics of Operations Research

Volumn 24, Issue 2, 1999, Pages 383-413

  Karger, David R ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; GRAPH THEORY; PROBABILITY; RANDOM PROCESSES; SAMPLING;

NETWORK DESIGN PROBLEMS; RANDOM SAMPLING;

PROBLEM SOLVING;

EID: 0032677628     PISSN: 0364765X     EISSN: None     Source Type: Journal    
DOI: 10.1287/moor.24.2.383     Document Type: Article

References (60)

1. Aggarwal, A. (Ed.). 1993. Proceedings of the 25th ACM Symposium on Theory of Computing, ACM Press.
2. _, N. Garg. 1994. A scaling technique for better network design. Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 233-240.
3. Agrawal, A., P. Klein, R. Ravi. 1995. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24(3) 440-456.
4. Ahuja, R. K., T. L. Magnanti, J. B. Orlin. 1993. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, New Jersey.
5. Aumann, Y., Y. Rabani. 1998. An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput. 27(1) 291-301.
6. 2) time. Proceedings of the 28th ACM Symposium on Theory of Computing, ACM Press, 47-55.
7. Chernoff, H. 1952. A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493-509.
8. Clarkson, K. L. 1987. New applications of random sampling in computational geometry. Discrete Comput. Geom. 2 195-222.
9. _. 1995. Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM 42(2) 488-499.
10. _, P. W. Shor. 1987. Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4(5) 387-421.
11. Dinitz, E. A., A. V. Karzanov, M. V. Lomonosov. 1976. On the structure of a family of minimum weighted cuts in a graph. Studies in Discrete Optimization. A. A. Fridman, ed. Nauka Publishers, 290-306.
12. Edmonds, J. 1965. Minimum partition of a matroid into independents subsets. J. Res. Nat. Bur. Standards 69 67-72.
13. Eppstein, D., Z. Galil, G. F. Italiano, A. Nissenzweig. 1992. Sparsification - A technique for speeding up dynamic graph algorithms. Proceedings of the 33rd Annual Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 60-69.
14. Erdös, P., A. Rényi. 1961. On the strength of connectedness of a random graph. Acta Math. Acad. Sci. Hungar. 12 261-267.
15. Eswaran, K. P., R. E. Tarjan. 1976. Augmentation problems. SIAM J. Comput. 5 653-665.
16. Feller, W. 1968. An Introduction to Probability Theory and Its Applications, Volume 1. 3rd ed. John Wiley and Sons, New York.
17. Floyd, R. W., R. L. Rivest. 1975. Expected time bounds for selection. Comm. ACM 18(3) 165-172.
18. Ford, Jr., L. R., D. R. Fulkerson. 1962. Flows in Networks. Princeton University Press. Princeton, New Jersey.
19. Frank, A. 1990. Packing paths, circuits, and cuts - A survey. Paths, Flows, and VLSI Layout, B. Korte, L. Lovász, H. J. Prömel, A. Schrijver, eds. Volume 9 of Algorithms and Combinatorics, Chapter 4. Springer-Verlag. Heidelberg.
20. Gabber, O., Z. Galil. 1981. Explicit construction of linear-sized superconcentrators. J. Comput. System Sci. 22 407-420.
21. Gabow, H. N. 1991. Applications of a poset representation to edge connectivity and graph rigidity. Proceedings of the 32nd Annual Symposium on the Foundations of Computer Science. IEEE: IEEE Computer Society Press, 812-821.
22. _. 1993. A framework for cost-scaling algorithms for submodular flow problems. Proceedings of the 34th Annual Symposium on the Foundations of Computer Science. IEEE Computer Society Press, 449-458.
23. Gabow, H. N. 1995. A matroid approach to finding edge connectivity and packing arborescences. J. Comput. System Sci. 50(2) 259-273.
24. _, M. X. Goemans, D. P. Williamson. 1993. An efficient approximation algorithm for the survivable network design problem. Proceedings of the Third MPS Conference on Integer Programming and Combinatorial Optimization, 57-74.
25. Goemans, M. X., D. J. Bertsimas. 1993. Survivable networks, linear programming relaxations and the parsimonious property. Math. Programming 60 145-166.
26. _, A. Goldberg, S. Plotkin, D. Shmoys, É. Tardos, D. Williamson. 1994. Improved approximation algorithms for network design problems. Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM. 223-232.
27. Grötschel, M., L. Lovász, A. Schrijver. 1988. Geometric Algorithms and Combinatorial Optimization, Volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, Germany.
28. Guibas, L. (Ed.) 1993. Proceedings of the 34th Annual Symposium on the Foundations of Computer Science. IEEE Computer Society Press.
29. Karger, D. R. 1994a. Random sampling in cut, flow, and network design problems. Proceedings of the 26th ACM Symposium on Theory of Computing, ACM Press, 648-657.
30. ~ karger.
31. _. 1994c, January. Using randomized sparsification to approximate minimum cuts. Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 424-432.
32. _. 1996, May. Minimum cuts in near-linear time. Proceedings of the 28th ACM Symposium on Theory of Computing, ACM Press, 56-63.
33. _. 1998a, January. Better random sampling algorithms for flows in undirected graphs. Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, H. Karloff, ed., 490-499.
34. _. 1998b, June. Random sampling and greedy sparsification in matroid optimization problems. Math. Programming B 82(1-2) 41-81.
35. _, P. N. Klein, R. E. Tarjan. 1995. A randomized linear-time algorithm to find minimum spanning trees. J. ACM 42(2) 321-328.
36. _, M. Levine. 1998, May. Finding maximum flows in simple undirected graphs seems faster than bipartite matching. Proceedings of the 29th ACM Symposium on Theory of Computing, ACM Press, 69-78.
37. 2) algorithm for minimum cuts. Proceedings of the 25th ACM Symposium on Theory of Computing. ACM Press, 757-765.
38. _, _. 1996, July. A new approach to the minimum cut problem. J. ACM 43(4) 601-640.
39. Khuller, S., B. Raghavachari. 1995, May. Improved approximation algorithms for uniform connectivity problems. Proceedings of the 27th ACM Symposium on Theory of Computing, ACM Press, 1-10.
40. _, B. Schieber. 1991, April. Efficient parallel algorithms for testing connectivity and finding disjoint s-t paths in graphs. SIAM J. Comput. 20(2) 352-375.
41. _, U. Vishkin. 1994, March. Biconnectivity approximations and graph carvings. J. ACM 41(2) 214-235.
42. Klein, P., S. A. Plotkin, C. Stein, É. Tardos. 1994. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. Comput. 23(3) 466-487.
43. Knuth, D. E. 1973. Fundamental Algorithms (2nd ed.), Volume 1 of The Art of Computer Programming. Addison-Wesley Publishing Company, Reading, MA.
44. Knuth, D. E., A. C. Yao. 1976. The complexity of nonuniform random number generation. Algorithms and Complexity: New Directions and Recent Results, J. F. Traub (ed.), Academic Press, New York 357-428.
45. Leighton, T., S. Rao. 1988, October. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. Proceedings of the 29th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 422-431.
46. Linial, N., E. London, Y. Rabinovich. 1995. The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2) 215-246.
47. Lomonosov, M. V., V. P. Polesskii. 1971. Lower bound of network reliability. Problems of Information Transmission 7 118-123.
48. Matula, D. W. 1993, January. A linear time 2 + ∉ approximation algorithm for edge connectivity. Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, 500-504. ACM-SIAM.
49. Miller, G. (Ed.) 1996, May. Proceedings of the 28th ACM Symposium on Theory of Computing. ACM Press.
50. _, P. Raghavan. 1995. Randomized Algorithms. Cambridge University Press, New York.
51. Nagamochi, H., T. Ibaraki. 1992a, February. Computing edge connectivity in multigraphs and capacitated graphs. SIAM J. Discrete Math. 5(1) 54-66.
52. _, _. 1992b. Linear time algorithms for finding k-edge connected and k-node connected spanning subgraphs. Algorithmica 7 583-596.
53. Nash-Williams, C. S. J. A. 1969. Well-balanced orientations of finite graphs and unobtrusive odd-vertex-pairings. Recent Progress in Combinatorics, W. T. Tutte (ed.), Academic Press, New York 133-149.
54. Raghavan, P. 1988, October. Probabilistic construction of deterministic algorithms: Approximate packing integer programs. J. Comput. System Sci. 37(2) 130-43.
55. _, C. D. Thompson. 1987. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4) 365-374.
56. Sleator, D. D. (Ed.) 1994, January. Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM.
57. Tarjan, R. E. 1983. Data Structures and Network Algorithms. Volume 44 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM.
58. Williamson, D., M. X. Goemans, M. Mihail, V. V. Vazirani. 1993, May. A primal-dual approximation algorithm for generalized Steiner problems. ACM Symposium on Theory of Computing, ACM Press, 708-717.
59. Winter, P. 1987. Generalized Steiner problem in outerplanar networks. Networks 17 129-167.
60. D. R. Karger: MIT Laboratory for Computer Science, Cambridge, Massachusetts 02139; e-mail: karger@lcs.mit.edu; URL: http://theory.lcs.mit.edu/~karger