Approximating minimum cuts under insertions

  • Monika Rauch Henzinger
Algorithms III
Part of the Lecture Notes in Computer Science book series (LNCS, volume 944)


This paper presents insertions-only algorithms for maintaining the exact and approximate size of the minimum edge cut and the minimum vertex cut of a graph. The algorithms output the approximate or exact size k in time O(1) or O(log n) and a cut of size k in time linear in its size. The amortized time per insertion is O(1/ε2) for a (2+ε)-approximation, O((log λ)((log n)/ε)2) for a (1+ε)-approximation, and O(λ log n) for the exact size of the minimum edge cut, where n is the number of nodes in the graph, λ is the size of the minimum cut and ε>0. The (2+ε)-approximation algorithm and the exact algorithm are deterministic, the (1+ε)-approximation algorithm is randomized. The algorithms are optimal in the sense that the time needed for m insertions matches the time needed by the best static algorithm on a m-edge graph. We also present a static 2-approximation algorithm for the size κ of the minimum vertex cut in a graph, which takes time O(n2min(√n,κ)). This is a factor of κ faster than the best algorithm for computing the exact size, which takes time O(κ2n23n1.5). We give an insertionsonly algorithm for maintaining a (2+ε)-approximation of the minimum vertex cut with amortized insertion time O(n(logκk)/ε).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Cheriyan, M. Y. Kao, and R. Thurimella, “Scan-first search and sparse certificates—an improved parallel algorithm for k-vertex connectivity”, SIAM Journal on Computing, 22, 1993, 157–174.Google Scholar
  2. 2.
    E. A. Dinitz, A. V. Karzanov, and M.V. Lomonosov, “On the structure of the system of minimal edge cuts in a graph”, Studies in Discrete Optimization, 1990, 290–306.Google Scholar
  3. 3.
    Ye. Dinitz, “Maintaining the 4-edge-connected components of a graph on-line”, Proc. 2nd Israel Symp. on Theory of Computing and Systems (ISTCS'93), IEEE Computing Society press, 1993, 88–97.Google Scholar
  4. 4.
    Ye. Dinitz, Z. Nutov, “A 2-level cactus model for the system of minimum and minimum+1 edge-cuts in a graph and its incremental maintenance”, to appear in Proc. 27nd Symp. on Theory of Computing, 1995.Google Scholar
  5. 5.
    D. Eppstein, Z. Galil, G. F. Italiano, A. Nissenzweig, “Sparsification — A Technique for Speeding up Dynamic Graph Algorithms” Proc. 33rd Symp. on Foundations of Computer Science, 1992, 60–69.Google Scholar
  6. 6.
    S. Even, “An algorithm for determining whether the connectivity of a graph is at least kSIAM Journal on Computing, 4, 1975, 393–396.Google Scholar
  7. 7.
    S. Even and R. E. Tarjan, “Network flow and testing graph connectivity”, SIAM Journal on Computing, 4, 1975, 507–518.Google Scholar
  8. 8.
    H. N. Gabow, “A matroid approach to finding edge connectivity and packing arborescences” Proc. 23rd Symp. on Theory of Computing, 1991, 112–122.Google Scholar
  9. 9.
    H. N. Gabow, “Applications of a poset representation to edge connectivity and graph rigidity” Proc. 32nd Symp. on Foundations of Computer Science, 1991, 812–821.Google Scholar
  10. 10.
    Z. Galil, “Finding the vertex connectivity of graphs”, SIAM Journal on Computing, 1980, 197–199.Google Scholar
  11. 11.
    Z. Galil and G. P. Italiano, “Maintaining the 3-edge-connected components of a graph on-line”, SIAM Journal on Computing, 1993, 11–28.Google Scholar
  12. 12.
    A. Ya. Gordon, “One algorithm for the solution of the minimax assignment problem”, Studies in Discrete Optimization, A. A. Fridman (Ed.), Nauka, Moscow, 1976, 327–333 (in Russian).Google Scholar
  13. 13.
    F. Harary, “Graph Theory”, Addison-Wesley, Reading, MA, 1969.Google Scholar
  14. 14.
    D. Karger, “Using randomized sparsification to approximate minimum cuts” Proc. 5th Symp. on Discrete Algorithms, 1994, 424–432.Google Scholar
  15. 15.
    D. Karger, “Random sampling in cut, flow, and network design problems”, Proc. 26rd Symp. on Theory of Computing, 1994, 648–657.Google Scholar
  16. 16.
    H. La Poutré, “Maintenance of 2-and 3-connected components of graphs, Part II: 2-and 3-edge-connected components and 2-vertex-connected components”, Tech.Rep. RUU-CS-90-27, Utrecht University, 1990.Google Scholar
  17. 17.
    D. W. Matula, “A linear time 2+ε approximation algorithm for edge connectivity” Proc. 4th Symp. on Discrete Algorithms, 1993, 500–504.Google Scholar
  18. 18.
    H. Nagamochi and T. Ibaraki, “Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph”, Algorithmica 7, 1992, 583–596.Google Scholar
  19. 19.
    D. D. Sleator, R. E. Tarjan, “A data structure for dynamic trees” J. Comput. System Sci. 24, 1983, 362–381.Google Scholar
  20. 20.
    J. Westbrook and R. E. Tarjan, “Maintaining bridge-connected and biconnected components on-line”, Algorithmica (7) 5/6, 1992, 433–464.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Monika Rauch Henzinger
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA

Personalised recommendations