Graph connectivity and its augmentation: Applications of MA orderings

Discrete Applied Mathematics

Volumn 123, Issue 1-3, 2002, Pages 447-472

  Nagamochi, Hiroshi ^a   Ibaraki, Toshihide ^b  

^a TOYOHASHI UNIVERSITY OF TECHNOLOGY   (Japan)

^b KYOTO UNIVERSITY   (Japan)

Author keywords

Edge connectivity; Graph augmentation; Graphs; MA ordering; Minimum cuts; Polynomial time algorithms

Indexed keywords

POLYNOMIAL APPROXIMATION; SURVEYS;

EDGE CONNECTIVITY; GRAPH AUGMENTATION; GRAPHS; MA ORDERINGS; MINIMUM CUT; POLYNOMIAL-TIME ALGORITHMS;

PROBLEM SOLVING;

EID: 84867958844     PISSN: 0166218X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0166-218X(01)00349-3     Document Type: Article

References (90)

1. R.K. Ahuja, T.L. Magnanti, and J.B. Orlin Network Flows: Theory, Algorithms, and Applications 1993 Prentice-Hall Englewood Cliffs, NJ
2. J. Bang-Jensen, A. Frank, and B. Jackson Preserving and increasing local edge-connectivity in mixed graphs SIAM J. Discrete Math. 8 1995 155 178
3. J. Bang-Jensen, H.N. Gabow, T. Jordán, and Z. Szigeti Edge-connectivity augmentation with partition constraints SIAM J. Discrete Math. 12 1999 160 207
4. J. Bang-Jensen, and T. Jordán Edge-connectivity augmentation preserving simplicity SIAM J. Discrete Math. 11 1998 603 623
5. A.A. Benczúr, Augmenting undirected connectivity in O (n 3) time, Proceedings of 26th ACM Symposium on Theory of Computing, 1994, pp. 658-667.
6. A.A. Benczúr, D.R. Karger, Augmenting undirected edge connectivity in O (n 2) time, Proceedings of 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 500-519.
7. G.-R. Cai, and Y.-G. Sun The minimum augmentation of any graph to k-edge-connected graph Networks 19 1989 151 172
8. C.S. Chekuri, A.V. Goldberg, D.R. Karger, M.S. Levine, C. Stein, Experimental study of minimum cut algorithms, Proceedings of 8th Annual ACM-SIAM Symposium on Discrete Algorithm, 1997, pp. 324-333.
9. E. Cheng, and T. Jordán Successive edge-connectivity augmentation problems Math. Program. Ser. B 84 1999 577 594
10. J. Cheriyan, M.-Y. Kao, and R. Thurimella Scan-first search and sparse certificates: an improved parallel algorithm for k-vertex connectivity SIAM J. Comput. 22 1993 157 174
11. J. Cheriyan, and R. Thurimella Fast algorithms for k-shredders and k-node connectivity augmentation J. Algorithms 33 1999 15 50
12. W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, and A. Schrijver Combin. Optimization 1998 Wiley-Interscience, Wiley, Inc New York
13. E.A. Dinits, A.V. Karzanov, M.V. Lomonosov, On the structure of a family of minimal weighted cuts in a graph, in: A.A. Fridman (Ed.), Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290-306 (in Russian).
14. S. Enni A note on mixed graphs and directed splitting off J. Graph Theory 27 1998 213 221
15. K.P. Eswaran, and R.E. Tarjan Augmentation problems SIAM J. Comput. 5 1976 653 665
16. S. Fialko, P. Mutzel, A new approximation algorithm for the planar augmentation problem, Proceedings of 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 260-269.
17. A. Frank Augmenting graphs to meet edge-connectivity requirements SIAM J. Discrete Math. 5 1992 25 53
18. A. Frank, Connectivity augmentation problems in network design, in: J.R. Birge, K.G. Murty (Ed.), Mathematical Programming: State of the Art 1994, The University of Michigan, Ann Arbor, MI 1994, pp. 34-63.
19. A. Frank, On the edge-connectivity algorithm of Nagamochi and Ibaraki, Laboratoire Artemis, IMAG, Université J. Fourier, Grenoble, March 1994.
20. A. Frank, T. Ibaraki, and H. Nagamochi On sparse subgraphs preserving connectivity properties J. Graph Theory 17 1993 275 281
21. A. Frank, and T. Jordán Minimal edge-coverings of pairs of sets J. Combin. Theory Ser. B 65 1995 73 110
22. A. Frank, and T. Jordán Directed vertex-connectivity augmentation Math. Program. Ser. B 84 1999 537 554
23. M.L. Fredman, and R.E. Tarjan Fibonacci heaps and their uses in improved network optimization algorithms J. ACM 34 1987 596 615
24. S. Fujishige Another simple proof of the validity of Nagamochi and Ibaraki's min-cut algorithm and Queyranne's extension to symmetric submodular function minimization J. Oper. Res. Soc. Japan 41 1998 626 628
25. H.N. Gabow, A matroid approach to finding edge connectivity and packing arborescences, Proceedings of 23rd ACM Symposium on Theory of Computing, New Orleans, Louisiana, 1991, pp. 112-122.
26. H.N. Gabow, Efficient splitting off algorithms for graphs, Proceedings of 26th ACM Symposium on Theory of Computing, 1994, 696-705.
27. H.N. Gabow, and T. Jordán How to make a square grid framework with cables rigid SIAM J. Comput. 30 2000 649 680
28. A.V. Goldberg, S. Rao, Flows in undirected unit capacity network, Proceedings of 38th Annual IEEE Symposium on Foundations of Computer Science, 1997, pp. 32-35.
29. D. Gusfield Optimal mixed graph augmentation SIAM J. Comput. 16 1987 599 612
30. J. Hao, and J.B. Orlin A faster algorithm for finding the minimum cut in a directed graph J. Algorithms 17 1994 424 446
31. M.R. Henzinger, S. Rao, and H.N. Gabow Computing vertex connectivity: new bounds from old techniques J. Algorithms 34 2000 222 250
32. T. Hsu On four-connecting a triconnected graph J. Algorithms 35 2000 202 234
33. T. Hsu, Undirected vertex-connectivity structure and smallest four-vertex connectivity augmentation, Lecture Notes in Computer Science 1004, Sixth International Symposium on Algorithms and Computation, Springer, Berlin, 1995, pp. 274-283.
34. T. Hsu, and M. Kao A unifying augmentation algorithm for two-edge-connectivity and biconnectivity J. Combin. Optim. 2 1998 237 256
35. T. Hsu, V. Ramachandran, A linear time algorithm for triconnectivity augmentation, Proceedings of 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 548-559.
36. T. Hsu, and V. Ramachandran Finding a smallest augmentation to biconnect a graph SIAM J. Comput. 22 1993 889 912
37. T.C. Hu Integer Programming and Network Flows 1969 Addison-Wesley Reading, MA
38. T. Ishii, Studies on multigraph connectivity augmentation problems, Ph.D. Thesis, Dept. of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan, 2000.
39. T. Ishii, H. Nagamochi, On the minimum augmentation of an ℓ-connected graph to a k-connected graph, Lecture Notes in Computer Science, 1851, Springer, Berlin, Seventh Biennial Scandinavian Workshop on Algorithm Theory, Bergen, Norway, July 5-7, 2000, pp. 286-299.
40. T. Ishii, H. Nagamochi, T. Ibaraki, Augmenting edge and vertex connectivities simultaneously, Lecture Notes in Computer Science, 1350, Springer, Berlin, Eighth International Symposium on Algorithms and Computation, 1997, pp. 102-111.
41. T. Ishii, H. Nagamochi, T. Ibaraki, Optimal augmentation of a biconnected graph to a k-edge-connected and triconnected graph, Proceedings of 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 280-289.
42. T. Ishii, H. Nagamochi, T. Ibaraki, k-edge and 3-vertex connectivity augmentation in an arbitrary multigraph, Lecture Notes in Computer Science, vol. 1533, Springer, Berlin, in: K.-Y. Chwa, O.H. Ibarra (Eds.), Algorithms and Computation, Ninth International Symposium on Algorithms and Computation, 1998, pp. 159-168.
43. T. Ishii, H. Nagamochi, T. Ibaraki, Augmenting a (k-1)-vertex-connected multigraph to an ℓ-edge-connected and k-vertex-connected multigraph, Lecture Notes in Computer Science, 1643, Springer, Berlin, Seventh Annual European Symposium on Algorithms, 1999, pp. 414-425.
44. T. Ishii, H. Nagamochi, and T. Ibaraki Optimal augmentation of a 2-vertex-connected multigraph to a k-edge-connected and 3-vertex-connected multigraph J. Combin. Optim. 4 2000 35 78
45. T. Ishii, H. Nagamochi, T. Ibaraki, Multigraph augmentation under biconnectivity and general edge-connectivity requirements, Networks 37 (2001) 144-155.
46. B. Jackson Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs J. Graph Theory 12 1988 429 436
47. B. Jackson, T. Jordán, A near optimal algorithm for vertex connectivity augmentation, in: Proc. 11th Annual Int. Symp. on Algorithms and Computation ISAAC'00, Lecture Notes in Computer Science, Vol. 1969, Springer, Berlin, 2000, pp. 326-337.
48. T. Jordán, Connectivity augmentation problem in graphs, Ph.D. Thesis, Department of Computer Science, Eötvös University, Budapest, Hungary, 1994.
49. T. Jordán On the optimal vertex-connectivity augmentation J. Combin. Theory Ser. B 63 1995 8 20
50. T. Jordán A note on the vertex-connectivity augmentation problem J. Combin. Theory Ser. B 71 1997 294 301
51. T. Jordán, Two NP-complete augmentation problems, Odense University Preprints #8, 1997.
52. T. Jordán, Edge-splitting problems with demands, Lecture Notes in Computer Science, 1610, Springer, Berlin, Seventh Conference on Integer Programming and Combinatorial Optimization, 1999, pp. 273-288.
53. G. Kant, Algorithms for Drawing Planar Graphs, Ph.D. Thesis, Dept. Of Computer Science, Utrecht University, 1993.
54. G. Kant Augmenting outerplanar graphs J. Algorithms 21 1996 1 25
55. G. Kant, H.L. Bodlaender, Planar graph augmentation problems, Lecture Notes in Computer Science, vol. 621, Springer, Berlin, 1992, pp. 258-271.
56. M. Kao Data security equals graph connectivity SIAM J. Discrete Math. 9 1996 87 100
57. D.R. Karger, Minimum cuts in near-linear time, Proceedings of 28th ACM Symposium on Theory of Computing, 1996, pp. 56-63.
58. D.R. Karger, M.S. Levine, Finding maximum flows in undirected graphs seems easier than bipartite matching, Proceedings of 30th ACM Symposium on Theory of Computing, 1998, pp. 69-78.
59. D.R. Karger, and R. Motwani An NC algorithm for minimum cuts SIAM J. Comput. 26 1997 255 272
60. D.R. Karger, C. Stein, An O (n 2) algorithm for minimum cuts, Proceedings of 25th ACM Symposium on Theory of Computing, 1993, pp. 757-765.
61. D.R. Karger, and C. Stein A new approach to the minimum cut problems J. ACM 43 1996 601 640
62. L. Lovász, Conference on Graph Theory, Prague, 1974.
63. L. Lovász Combinatorial Problems and Exercises 1979 North-Holland Amsterdam
64. W. Mader A reduction method for edge-connectivity in graphs Ann. Discrete Math. 3 1978 145 164
65. W. Mader Konstruktion aller n-fach kantenzusammenhängenden Digraphen European J. Combin. 3 1982 63 67
66. D.W. Matula, A linear time 2+ε approximation algorithm for edge connectivity, Proceedings of 4th Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 500-504.
67. D.W. Matula, Two results on search and edge connectivity, Graph Algorithms and Applications, Dagtuhl-Seminar-Report, vol. 145, 1996, p. 16.
68. H. Nagamochi, P. Eades, Edge-splitting and edge-connectivity augmentation in planar graphs, Lecture Notes in Computer Science, 1412, Springer, Berlin, in: R.E. Bixby, E.A. Boyd, R.Z. Ros-Mercado (Eds.), Sixth Conference on Integer Programming and Combinatorial Optimization, 1998, pp. 96-111.
69. H. Nagamochi, and T. Ibaraki A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph Algorithmica 7 1992 583 596
70. H. Nagamochi, and T. Ibaraki Computing edge-connectivity in multigraphs and capacitated graphs SIAM J. Discrete Mathematics 5 1992 54 66
71. H. Nagamochi, and T. Ibaraki Deterministic O (nm) time edge-splitting in undirected graphs J. Combin. Optim. 1 1997 5 46
72. H. Nagamochi, and T. Ibaraki A note on minimizing submodular functions Inform. Process. Lett. 67 1998 239 244
73. H. Nagamochi, T. Ibaraki, Polyhedral structure of submodular and posi-modular systems, Lecture Notes in Computer Science, 1533, in: K.-Y. Chwa, O.H. Ibarra (Eds.), Springer, Berlin, Algorithms and Computation, Ninth International Symposium on Algorithms and Computations, 1998, pp. 169-178.
74. H. Nagamochi, and T. Ibaraki Augmenting edge-connectivity over the entire range in O (nm) time J. Algorithms 30 1999 253 301
75. H. Nagamochi, T. Ibaraki, Polyhedral structure of submodular and posi-modular systems, Discrete Appl. Math. 107 (2000) 165-189.
76. H. Nagamochi, T. Ono, and T. Ibaraki Implementing an efficient minimum capacity cut algorithm Math. Program. 67 1994 325 341
77. H. Nagamochi, T. Ishii, and T. Ibaraki A simple and constructive proof of a minimum cut algorithm Inst. Electron. Inform. Comm. Eng. Trans. Fundamentals E82-A 1999 2231 2236
78. H. Nagamochi, S. Nakamura, and T. Ibaraki A simplified O (nm) time edge-splitting algorithm in undirected graphs Algorithmica 26 2000 56 67
79. H. Nagamochi, Y. Nakao, and T. Ibaraki A fast algorithm for cactus representations of minimum cuts J. Japan Soc. Ind. Appl. Math. 17 2000 245 264
80. M. Padberg, and G. Rinaldi An efficient algorithm for the minimum capacity cut problem Math. Program. 47 1990 19 36
81. J.C. Picard, and M. Queyranne On the structure of all minimum cuts in a network and applications Math. Program. Study 13 1980 8 16
82. J. Plesnik, Minimum block containing a given graph, Archiv der Mathmatik, XXVII 1976, Fasc. 6, pp. 668-672.
83. M. Queyranne Minimizing symmetric submodular functions Math. Program. 82 1998 3 12
84. S. Raghavan, and T.L. Magnanti Network Connectivity M. Dell'Amico, F. Maffioli, S. Martello, Annotated Bibliographies in Combinatorial Optimization 1997 Wiley New York
85. R. Rizzi On minimizing symmetric set functions Combinatorica 20 2000 445 450
86. M. Stoer, and F. Wagner A simple min-cut algorithm J. ACM 44 1997 585 591
87. R.E. Tarjan, and M. Yannakakis Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs SIAM J. Comput. 13 1984 566 579
88. S. Tsukiyama, K. Koike, I. Shirakawa, An algorithm to eliminate all complex triangles in a maximal planar graph for use in VLSI floor-plan, Proceedings of ISCAS'86, 1986, pp. 321-324.
89. T. Watanabe, and A. Nakamura Edge-connectivity augmentation problems J. Comput. System Sci. 35 1987 96 144
90. T. Watanabe, and A. Nakamura A minimum 3-connectivity augmentation of a graph J. Comput. System Sci. 46 1993 91 128