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A Fast Algorithm for Computing Minimum 3-Way and 4-Way Cuts

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Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

Abstract

For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k = 3, 4. The algorithm runs in O(n k−2(nF(n,m) + C 2(n,m) + n 2)) = O(mn k log(n 2/m)) time for k = 3, 4, where F(n,m) and C 2(n,m) denote respectively the time bounds required to solve the maximum flow problem and the minimum 2-way cut problem in G. The bound for k = 3 matches the current best deterministic bound Õ (mn 3) for weighted graphs, but improves the bound Õ(mn 3) to O(n(nF(n,m) + C 2(n,m) + n 2)) = O(min{mn 8/3,m 3/2 n 2}) for unweighted graphs. The bound Õ(mn 4) for k = 4 improves the previous best randomized bound Õ(n 6) (for m = o(n 2)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system.

This research was partially supported by the Scientific Grant-in-Aid from Ministry of Education, Science, Sports and Culture of Japan, and the subsidy from the Inamori Foundation.

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References

  1. M. Burlet and O. Goldschmidt, A new and improved algorithm for the 3-cut problem, Operations Research Letters, vol.21, (1997), pp. 225–227.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Even and R. E. Tarjan, Network flow and testing graph connectivity, SIAM J. Computing, vol.4, (1975), pp. 507–518.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. V. Goldberg and S. Rao, Beyond the flow decomposition barrier, Proc. 38th IEEE Annual Symp. on Foundations of Computer Science, (1997), pp. 2–11.

    Google Scholar 

  4. A. V. Goldberg and R. E. Tarjan, A new approach to the maximum flow problem, J. ACM, vol.35, (1988), pp. 921–940.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin (1988).

    MATH  Google Scholar 

  6. O. Goldschmidt and D. S. Hochbaum, A polynomial algorithm for the k-cut problem for fixed k, Mathematics of Operations Research, vol.19, (1994), pp. 24–37.

    MATH  MathSciNet  Google Scholar 

  7. J. Hao and J. Orlin, A faster algorithm for finding the minimum cut in a directed graph, J. Algorithms, vol. 17, (1994), pp. 424–446.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. R. Henzinger, P. Klein, S. Rao and D. Williamson, Faster shortest-path algorithms for planar graphs, J. Comp. Syst. Sc., vol. 53, (1997), pp. 2–23.

    Google Scholar 

  9. D. S. Hochbaum and D. B. Shmoys, An O(|V|2) algorithm for the planar 3-cut problem, SIAM J. Algebraic Discrete Methods, vol.6, (1985), pp. 707–712.

    Article  MATH  MathSciNet  Google Scholar 

  10. Y. Kamidoi, S. Wakabayashi and N. Yoshida, Faster algorithms for finding a minimum k-way cut in a weighted graph, Proc. IEEE International Symposium on Circuits and Systems, (1997), pp. 1009–1012.

    Google Scholar 

  11. Y. Kamidoi, S. Wakabayashi and N. Yoshida, A new approach to the minimum k-way partition problem for weighted graphs, Technical Report of Inst. Electron. Inform. Comm. Eng., COMP97-25, (1997), pp. 25–32.

    Google Scholar 

  12. S. Kapoor, On minimum 3-cuts and approximating k-cuts using cut trees, Lecture Notes in Computer Science 1084, Springer-Verlag, (1996), pp. 132–146.

    Google Scholar 

  13. D. R. Karger, Minimum cuts in near-linear time, Proceedings 28th ACM Symposium on Theory of Computing, (1996), pp. 56–63.

    Google Scholar 

  14. D. R. Karger and C. Stein, An Õ(n 2) algorithm for minimum cuts, Proceedings 25th ACM Symposium on Theory of Computing, (1993), pp. 757–765.

    Google Scholar 

  15. D. R. Karger and C. Stein, A new approach to the minimum cut problems, J. ACM, vol.43, no.4, (1996), pp. 601–640.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. V. Karzanov, O nakhozhdenii maksimal’nogo potoka v setyakh spetsial’nogo vida i nekotorykh prilozheniyakh, In Mathematicheskie Voprosy Upravleniya Proizvodstvom, volume 5, Moscow State University Press, Moscow, (1973). In Russian; title translation: On finding maximum flows in networks with special structure and some applications.

    Google Scholar 

  17. N. Katoh, T. Ibaraki and H. Mine, An efficient algorithm for K shortest simple paths, Networks, vol.12, (1982), pp. 441–427.

    Article  MathSciNet  Google Scholar 

  18. R. Klimmek and F. Wagner, A simple hypergraph min cut algorithm, Technical Report B 96-02, Department of Mathematics and Computer Science, Freie Universität Berlin (1996).

    Google Scholar 

  19. E. L. Lawler, Cutsets and partitions of hypergraphs, Networks, vol.3, (1973), pp. 275–285.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. H. Lee, M. Kim and C. I. Park, An efficient k-way graph partitioning algorithm for task allocation in parallel computing systems, Proc. IEEE Int. Conf. on Computer-Aided Design, (1990), pp. 748–751.

    Google Scholar 

  21. T. Lengaur, Combinatorial Algorithms for Integrated Circuit Layout, Wiley (1990).

    Google Scholar 

  22. H. Nagamochi and T. Ibaraki, Computing the edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics, vol.5, (1992), pp. 54–66.

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Nagamochi and T. Ibaraki, A note on minimizing submodular functions, Information Processing Letters, vol. 67, (1998), pp.239–244.

    Article  MathSciNet  Google Scholar 

  24. H. Nagamochi, K. Nishimura and T. Ibaraki, Computing all small cuts in undirected networks, SIAM J. Discrete Mathematics, vol. 10, (1997), pp. 469–481.

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Nagamochi, S. Katayama and T. Ibaraki, A faster algorithm for computing minimum 5-way and 6-way cuts in graphs, Technical Report, Department of Applied Mathematics and Physics, Kyoto University, 1999.

    Google Scholar 

  26. Pulleyblank, Presentation at SIAM Meeting on Optimization, MIT, Boston, MA (1982).

    Google Scholar 

  27. M. Queyranne, Minimizing symmetric submodular functions, Mathematical Programming, 82 (1998), pp. 3–12.

    MathSciNet  Google Scholar 

  28. V. V. Vazirani and M. Yannakakis, Suboptimal cuts: Their enumeration, weight, and number, Lecture Notes in Computer Science, 623, Springer-Verlag, (1992), pp. 366–377.

    Google Scholar 

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Authors and Affiliations

  1. Kyoto University, Kyoto, Japan, 606-8501

    Hiroshi Nagamochi & Toshihide Ibaraki

Authors
  1. Hiroshi Nagamochi
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  2. Toshihide Ibaraki
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Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Nagamochi, H., Ibaraki, T. (1999). A Fast Algorithm for Computing Minimum 3-Way and 4-Way Cuts. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_28

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  • DOI: https://doi.org/10.1007/3-540-48777-8_28

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

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