Efficient Algorithms for the k Smallest Cuts Enumeration

  • Li-Pu Yeh
  • Biing-Feng Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


In this paper, we study the problems of enumerating cuts of a graph by non-decreasing weights. There are four problems, depending on whether the graph is directed or undirected, and on whether we consider all cuts of the graph or only s-t cuts for a given pair of vertices s, t. Efficient algorithms for these problems with \({\tilde{\mbox{O}}}(n^2m)\) delay between two successive outputs have been known since 1992, due to Vazirani and Yannakakis. In this paper, improved algorithms are presented. The delays of the presented algorithms are O(nm log(n 2/m)).


Undirected Graph Basic Partition Sink Vertex Sink Side Basic Subroutine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Burlet, M., Goldschmidt, O.: A New and Improved Algorithm for the 3-Cut Problem. Operations Research Letters 21, 225–227 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chekuri, C.S., Goldberg, A.V., Karger, D.R., Levine, M.S., Stein, C.: Experimental Study of Minimum Cut Algorithms. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 324–333 (1997)Google Scholar
  4. 4.
    Dinits, E.A., Karzanov, A.V., Lomonosov, M.V.: On the Structure of a Family of Minimal Weighted Cuts in a Graph. In: Fridman, A.A. (ed.) Studies in Discrete Optimization, Nauka, Moscow, pp. 290–306 (1976)Google Scholar
  5. 5.
    Fleischer, L.: Building Chain and Cactus Representations of All Minimum Cuts from Hao-Orlin in the Same Asymptotic Run Time. Journal of Algorithms 33, 51–72 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goldberg, A.V., Tarjan, R.E.: A New Approach to the Maximum Flow Problem. Journal of the ACM 35, 921–940 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Goldschmidt, O., Hochbaum, D.S.: Polynomial Algorithm for the k-Cut Problem for Fixed k. Mathematics of Operation Research 19, 24–37 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gomory, R.E., Hu, T.C.: Multi-terminal Network Flows. Journal of the Society for Industrial and Applied Mathematics 9, 551–570 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hao, J., Orlin, J.B.: A Faster Algorithm for Finding the Minimum Cut in a Directed Graph. Journal of Algorithms 17, 424–446 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kamidoi, Y., Wakabayashi, S., Yoshida, N.: A Divide-and-Conquer Approach to the Minimum k-Way Cut Problem. Algorithmica 32, 262–276 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kamidoi, Y., Yoshida, N., Nagamochi, H.: A Deterministic Algorithm for Finding all Minimum k-Way Cuts. SIAM Journal on Computing 36, 1315–1327 (2006)MathSciNetGoogle Scholar
  12. 12.
    Kapoor, S.: On Minimum 3-Cuts and Approximating k-Cuts Using Cut Trees. In: Proceedings of the 5th Integer Programming and Combinatorial Optimization Conference, pp. 132–146 (1996)Google Scholar
  13. 13.
    Karger, D.R., Stein, C.: A New Approach to the Minimum Cut Problem. Journal of the ACM 43, 601–640 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Karger, D.R.: A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem. SIAM Journal on Computing 29, 492–514 (1999)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Karger, D.R.: Minimum Cuts in Near-Linear Time. Journal of the ACM 47, 46–76 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Karzanov, A.V., Timofeev, E.A.: Efficient algorithms for Finding All Minimal Edge Cuts of a Nonoriented Graph. Cybernetics 22, 156–162 (1986)zbMATHCrossRefGoogle Scholar
  17. 17.
    King, V., Rao, S., Tarjan, R.E.: A Faster Deterministic Maximum Flow Algorithm. Journal of Algorithms 17, 447–474 (1994)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Levine, M.S.: Faster Randomized Algorithms for Computing Minimum 3, 4, 5, 6-way Cuts. In: Proceedings of the Eleventh ACM-SIAM Symposium on Discrete Algorithms, pp. 735–742 (2000)Google Scholar
  19. 19.
    Nagamochi, H., Ibaraki, T.: Computing the Edge-Connectivity of Multigraphs and Capacitated Graphs. SIAM Journal on Discrete Mathematics 5, 54–66 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nagamochi, H., Ibaraki, T.: A Fast Algorithm for Computing Minimum 3-way and 4-way Cuts. Mathematical Programming 88, 507–520 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Nagamochi, H., Nishimura, K., Ibaraki, T.: Computing all Small Cuts in Undirected Networks. SIAM Journal on Discrete Mathematics 10, 469–481 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Nagamochi, H., Nishimura, K., Ibaraki, T.: A Faster Algorithm for Computing Minimum 5-Way and 6-Way Cuts in Graphs. Journal of Combinatorial Optimization 4, 151–169 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Picard, J.C., Queyrane, M.: On the Structure of All Minimum Cuts in a Network and Applications. Mathematical Programming Study 13, 8–16 (1980)zbMATHGoogle Scholar
  24. 24.
    Vazirani, V., Yannakakis, M.: Suboptimal Cuts: Their Enumeration, Weight and Number. In: Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pp. 366–377 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Li-Pu Yeh
    • 1
  • Biing-Feng Wang
    • 1
  1. 1.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan

Personalised recommendations